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Metamath Proof Explorer

Theorem updjudhcoinlf

Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022)

Ref Expression
Hypotheses updjud.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )
updjud.g ⊢ ( 𝜑 → 𝐺 : 𝐵 ⟶ 𝐶 )
updjudhf.h ⊢ 𝐻 = ( 𝑥 ∈ ( 𝐴 ⊔ 𝐵 ) ↦ if ( ( 1st ‘ 𝑥 ) = ∅ , ( 𝐹 ‘ ( 2nd ‘ 𝑥 ) ) , ( 𝐺 ‘ ( 2nd ‘ 𝑥 ) ) ) )
Assertion updjudhcoinlf ( 𝜑 → ( 𝐻 ∘ ( inl ↾ 𝐴 ) ) = 𝐹 )

Proof

Step Hyp Ref Expression
1 updjud.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )
2 updjud.g ⊢ ( 𝜑 → 𝐺 : 𝐵 ⟶ 𝐶 )
3 updjudhf.h ⊢ 𝐻 = ( 𝑥 ∈ ( 𝐴 ⊔ 𝐵 ) ↦ if ( ( 1st ‘ 𝑥 ) = ∅ , ( 𝐹 ‘ ( 2nd ‘ 𝑥 ) ) , ( 𝐺 ‘ ( 2nd ‘ 𝑥 ) ) ) )
4 1 2 3 updjudhf ⊢ ( 𝜑 → 𝐻 : ( 𝐴 ⊔ 𝐵 ) ⟶ 𝐶 )
5 4 ffnd ⊢ ( 𝜑 → 𝐻 Fn ( 𝐴 ⊔ 𝐵 ) )
6 inlresf ⊢ ( inl ↾ 𝐴 ) : 𝐴 ⟶ ( 𝐴 ⊔ 𝐵 )
7 ffn ⊢ ( ( inl ↾ 𝐴 ) : 𝐴 ⟶ ( 𝐴 ⊔ 𝐵 ) → ( inl ↾ 𝐴 ) Fn 𝐴 )
8 6 7 mp1i ⊢ ( 𝜑 → ( inl ↾ 𝐴 ) Fn 𝐴 )
9 frn ⊢ ( ( inl ↾ 𝐴 ) : 𝐴 ⟶ ( 𝐴 ⊔ 𝐵 ) → ran ( inl ↾ 𝐴 ) ⊆ ( 𝐴 ⊔ 𝐵 ) )
10 6 9 mp1i ⊢ ( 𝜑 → ran ( inl ↾ 𝐴 ) ⊆ ( 𝐴 ⊔ 𝐵 ) )
11 fnco ⊢ ( ( 𝐻 Fn ( 𝐴 ⊔ 𝐵 ) ∧ ( inl ↾ 𝐴 ) Fn 𝐴 ∧ ran ( inl ↾ 𝐴 ) ⊆ ( 𝐴 ⊔ 𝐵 ) ) → ( 𝐻 ∘ ( inl ↾ 𝐴 ) ) Fn 𝐴 )
12 5 8 10 11 syl3anc ⊢ ( 𝜑 → ( 𝐻 ∘ ( inl ↾ 𝐴 ) ) Fn 𝐴 )
13 1 ffnd ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
14 fvco2 ⊢ ( ( ( inl ↾ 𝐴 ) Fn 𝐴 ∧ 𝑎 ∈ 𝐴 ) → ( ( 𝐻 ∘ ( inl ↾ 𝐴 ) ) ‘ 𝑎 ) = ( 𝐻 ‘ ( ( inl ↾ 𝐴 ) ‘ 𝑎 ) ) )
15 8 14 sylan ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( ( 𝐻 ∘ ( inl ↾ 𝐴 ) ) ‘ 𝑎 ) = ( 𝐻 ‘ ( ( inl ↾ 𝐴 ) ‘ 𝑎 ) ) )
16 fvres ⊢ ( 𝑎 ∈ 𝐴 → ( ( inl ↾ 𝐴 ) ‘ 𝑎 ) = ( inl ‘ 𝑎 ) )
17 16 adantl ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( ( inl ↾ 𝐴 ) ‘ 𝑎 ) = ( inl ‘ 𝑎 ) )
18 17 fveq2d ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐻 ‘ ( ( inl ↾ 𝐴 ) ‘ 𝑎 ) ) = ( 𝐻 ‘ ( inl ‘ 𝑎 ) ) )
19 fveqeq2 ⊢ ( 𝑥 = ( inl ‘ 𝑎 ) → ( ( 1st ‘ 𝑥 ) = ∅ ↔ ( 1st ‘ ( inl ‘ 𝑎 ) ) = ∅ ) )
20 2fveq3 ⊢ ( 𝑥 = ( inl ‘ 𝑎 ) → ( 𝐹 ‘ ( 2nd ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) )
21 2fveq3 ⊢ ( 𝑥 = ( inl ‘ 𝑎 ) → ( 𝐺 ‘ ( 2nd ‘ 𝑥 ) ) = ( 𝐺 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) )
22 19 20 21 ifbieq12d ⊢ ( 𝑥 = ( inl ‘ 𝑎 ) → if ( ( 1st ‘ 𝑥 ) = ∅ , ( 𝐹 ‘ ( 2nd ‘ 𝑥 ) ) , ( 𝐺 ‘ ( 2nd ‘ 𝑥 ) ) ) = if ( ( 1st ‘ ( inl ‘ 𝑎 ) ) = ∅ , ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) , ( 𝐺 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) ) )
23 22 adantl ⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑥 = ( inl ‘ 𝑎 ) ) → if ( ( 1st ‘ 𝑥 ) = ∅ , ( 𝐹 ‘ ( 2nd ‘ 𝑥 ) ) , ( 𝐺 ‘ ( 2nd ‘ 𝑥 ) ) ) = if ( ( 1st ‘ ( inl ‘ 𝑎 ) ) = ∅ , ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) , ( 𝐺 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) ) )
24 1stinl ⊢ ( 𝑎 ∈ 𝐴 → ( 1st ‘ ( inl ‘ 𝑎 ) ) = ∅ )
25 24 adantl ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 1st ‘ ( inl ‘ 𝑎 ) ) = ∅ )
26 25 adantr ⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑥 = ( inl ‘ 𝑎 ) ) → ( 1st ‘ ( inl ‘ 𝑎 ) ) = ∅ )
27 26 iftrued ⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑥 = ( inl ‘ 𝑎 ) ) → if ( ( 1st ‘ ( inl ‘ 𝑎 ) ) = ∅ , ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) , ( 𝐺 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) ) = ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) )
28 23 27 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑥 = ( inl ‘ 𝑎 ) ) → if ( ( 1st ‘ 𝑥 ) = ∅ , ( 𝐹 ‘ ( 2nd ‘ 𝑥 ) ) , ( 𝐺 ‘ ( 2nd ‘ 𝑥 ) ) ) = ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) )
29 djulcl ⊢ ( 𝑎 ∈ 𝐴 → ( inl ‘ 𝑎 ) ∈ ( 𝐴 ⊔ 𝐵 ) )
30 29 adantl ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( inl ‘ 𝑎 ) ∈ ( 𝐴 ⊔ 𝐵 ) )
31 1 adantr ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝐹 : 𝐴 ⟶ 𝐶 )
32 2ndinl ⊢ ( 𝑎 ∈ 𝐴 → ( 2nd ‘ ( inl ‘ 𝑎 ) ) = 𝑎 )
33 32 adantl ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 2nd ‘ ( inl ‘ 𝑎 ) ) = 𝑎 )
34 simpr ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ∈ 𝐴 )
35 33 34 eqeltrd ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 2nd ‘ ( inl ‘ 𝑎 ) ) ∈ 𝐴 )
36 31 35 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) ∈ 𝐶 )
37 3 28 30 36 fvmptd2 ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐻 ‘ ( inl ‘ 𝑎 ) ) = ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) )
38 18 37 eqtrd ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐻 ‘ ( ( inl ↾ 𝐴 ) ‘ 𝑎 ) ) = ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) )
39 33 fveq2d ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐹 ‘ ( 2nd ‘ ( inl ‘ 𝑎 ) ) ) = ( 𝐹 ‘ 𝑎 ) )
40 15 38 39 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( ( 𝐻 ∘ ( inl ↾ 𝐴 ) ) ‘ 𝑎 ) = ( 𝐹 ‘ 𝑎 ) )
41 12 13 40 eqfnfvd ⊢ ( 𝜑 → ( 𝐻 ∘ ( inl ↾ 𝐴 ) ) = 𝐹 )