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Metamath Proof Explorer

Theorem abrexco

Description: Composition of two image maps C ( y ) and B ( w ) . (Contributed by NM, 27-May-2013)

Ref Expression
Hypotheses abrexco.1 ⊢ 𝐵 ∈ V
abrexco.2 ⊢ ( 𝑦 = 𝐵 → 𝐶 = 𝐷 )
Assertion abrexco { 𝑥 ∣ ∃ 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } 𝑥 = 𝐶 } = { 𝑥 ∣ ∃ 𝑤 ∈ 𝐴 𝑥 = 𝐷 }

Proof

Step Hyp Ref Expression
1 abrexco.1 ⊢ 𝐵 ∈ V
2 abrexco.2 ⊢ ( 𝑦 = 𝐵 → 𝐶 = 𝐷 )
3 df-rex ⊢ ( ∃ 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } 𝑥 = 𝐶 ↔ ∃ 𝑦 ( 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } ∧ 𝑥 = 𝐶 ) )
4 vex ⊢ 𝑦 ∈ V
5 eqeq1 ⊢ ( 𝑧 = 𝑦 → ( 𝑧 = 𝐵 ↔ 𝑦 = 𝐵 ) )
6 5 rexbidv ⊢ ( 𝑧 = 𝑦 → ( ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 ↔ ∃ 𝑤 ∈ 𝐴 𝑦 = 𝐵 ) )
7 4 6 elab ⊢ ( 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } ↔ ∃ 𝑤 ∈ 𝐴 𝑦 = 𝐵 )
8 7 anbi1i ⊢ ( ( 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } ∧ 𝑥 = 𝐶 ) ↔ ( ∃ 𝑤 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) )
9 r19.41v ⊢ ( ∃ 𝑤 ∈ 𝐴 ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) ↔ ( ∃ 𝑤 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) )
10 8 9 bitr4i ⊢ ( ( 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } ∧ 𝑥 = 𝐶 ) ↔ ∃ 𝑤 ∈ 𝐴 ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) )
11 10 exbii ⊢ ( ∃ 𝑦 ( 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } ∧ 𝑥 = 𝐶 ) ↔ ∃ 𝑦 ∃ 𝑤 ∈ 𝐴 ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) )
12 3 11 bitri ⊢ ( ∃ 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } 𝑥 = 𝐶 ↔ ∃ 𝑦 ∃ 𝑤 ∈ 𝐴 ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) )
13 rexcom4 ⊢ ( ∃ 𝑤 ∈ 𝐴 ∃ 𝑦 ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) ↔ ∃ 𝑦 ∃ 𝑤 ∈ 𝐴 ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) )
14 12 13 bitr4i ⊢ ( ∃ 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } 𝑥 = 𝐶 ↔ ∃ 𝑤 ∈ 𝐴 ∃ 𝑦 ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) )
15 2 eqeq2d ⊢ ( 𝑦 = 𝐵 → ( 𝑥 = 𝐶 ↔ 𝑥 = 𝐷 ) )
16 1 15 ceqsexv ⊢ ( ∃ 𝑦 ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) ↔ 𝑥 = 𝐷 )
17 16 rexbii ⊢ ( ∃ 𝑤 ∈ 𝐴 ∃ 𝑦 ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐶 ) ↔ ∃ 𝑤 ∈ 𝐴 𝑥 = 𝐷 )
18 14 17 bitri ⊢ ( ∃ 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } 𝑥 = 𝐶 ↔ ∃ 𝑤 ∈ 𝐴 𝑥 = 𝐷 )
19 18 abbii ⊢ { 𝑥 ∣ ∃ 𝑦 ∈ { 𝑧 ∣ ∃ 𝑤 ∈ 𝐴 𝑧 = 𝐵 } 𝑥 = 𝐶 } = { 𝑥 ∣ ∃ 𝑤 ∈ 𝐴 𝑥 = 𝐷 }