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Metamath Proof Explorer

Theorem bnj1326

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1326.1 ⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
bnj1326.2 ⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1326.3 ⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
bnj1326.4 ⊢ 𝐷 = ( dom 𝑔 ∩ dom ℎ )
Assertion bnj1326 ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ) → ( 𝑔 ↾ 𝐷 ) = ( ℎ ↾ 𝐷 ) )

Proof

Step Hyp Ref Expression
1 bnj1326.1 ⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2 bnj1326.2 ⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3 bnj1326.3 ⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4 bnj1326.4 ⊢ 𝐷 = ( dom 𝑔 ∩ dom ℎ )
5 eleq1w ⊢ ( 𝑞 = ℎ → ( 𝑞 ∈ 𝐶 ↔ ℎ ∈ 𝐶 ) )
6 5 3anbi3d ⊢ ( 𝑞 = ℎ → ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ) ) )
7 dmeq ⊢ ( 𝑞 = ℎ → dom 𝑞 = dom ℎ )
8 7 ineq2d ⊢ ( 𝑞 = ℎ → ( dom 𝑔 ∩ dom 𝑞 ) = ( dom 𝑔 ∩ dom ℎ ) )
9 8 reseq2d ⊢ ( 𝑞 = ℎ → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
10 4 reseq2i ⊢ ( 𝑔 ↾ 𝐷 ) = ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) )
11 9 10 eqtr4di ⊢ ( 𝑞 = ℎ → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑔 ↾ 𝐷 ) )
12 8 reseq2d ⊢ ( 𝑞 = ℎ → ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
13 reseq1 ⊢ ( 𝑞 = ℎ → ( 𝑞 ↾ ( dom 𝑔 ∩ dom ℎ ) ) = ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
14 12 13 eqtrd ⊢ ( 𝑞 = ℎ → ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
15 4 reseq2i ⊢ ( ℎ ↾ 𝐷 ) = ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) )
16 14 15 eqtr4di ⊢ ( 𝑞 = ℎ → ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( ℎ ↾ 𝐷 ) )
17 11 16 eqeq12d ⊢ ( 𝑞 = ℎ → ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) ↔ ( 𝑔 ↾ 𝐷 ) = ( ℎ ↾ 𝐷 ) ) )
18 6 17 imbi12d ⊢ ( 𝑞 = ℎ → ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ) → ( 𝑔 ↾ 𝐷 ) = ( ℎ ↾ 𝐷 ) ) ) )
19 eleq1w ⊢ ( 𝑝 = 𝑔 → ( 𝑝 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶 ) )
20 19 3anbi2d ⊢ ( 𝑝 = 𝑔 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) ) )
21 dmeq ⊢ ( 𝑝 = 𝑔 → dom 𝑝 = dom 𝑔 )
22 21 ineq1d ⊢ ( 𝑝 = 𝑔 → ( dom 𝑝 ∩ dom 𝑞 ) = ( dom 𝑔 ∩ dom 𝑞 ) )
23 22 reseq2d ⊢ ( 𝑝 = 𝑔 → ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑝 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
24 reseq1 ⊢ ( 𝑝 = 𝑔 → ( 𝑝 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
25 23 24 eqtrd ⊢ ( 𝑝 = 𝑔 → ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
26 22 reseq2d ⊢ ( 𝑝 = 𝑔 → ( 𝑞 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
27 25 26 eqeq12d ⊢ ( 𝑝 = 𝑔 → ( ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) ↔ ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) ) )
28 20 27 imbi12d ⊢ ( 𝑝 = 𝑔 → ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) ) ) )
29 eqid ⊢ ( dom 𝑝 ∩ dom 𝑞 ) = ( dom 𝑝 ∩ dom 𝑞 )
30 1 2 3 29 bnj1311 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) )
31 28 30 chvarvv ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
32 18 31 chvarvv ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ) → ( 𝑔 ↾ 𝐷 ) = ( ℎ ↾ 𝐷 ) )