Theorem elima3 2608

Description: Membership in an image. Theorem 34 of [Suppes] p. 65.

Hypothesis

Ref	Expression
elima.1	⊢ A ∈ V

Assertion

Ref	Expression
elima3	⊢ (A ∈ (B “ C) ↔ ∃x(x ∈ C ∧ ⟨x, A⟩ ∈ B))

Distinct variable group(s):   x,A   x,B   x,C

Proof of Theorem elima3

Step	Hyp	Ref	Expression
1		elima.1	. . 3 ⊢ A ∈ V
2	1	elima2 2607	. 2 ⊢ (A ∈ (B “ C) ↔ ∃x(x ∈ C ∧ xBA))
3		df-br 2063	. . . 4 ⊢ (xBA ↔ ⟨x, A⟩ ∈ B)
4	3	anbi2i 367	. . 3 ⊢ ((x ∈ C ∧ xBA) ↔ (x ∈ C ∧ ⟨x, A⟩ ∈ B))
5	4	biex 733	. 2 ⊢ (∃x(x ∈ C ∧ xBA) ↔ ∃x(x ∈ C ∧ ⟨x, A⟩ ∈ B))
6	2, 5	bitr 151	1 ⊢ (A ∈ (B “ C) ↔ ∃x(x ∈ C ∧ ⟨x, A⟩ ∈ B))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054   “ cima 2413

This theorem is referenced by:  hbima 2609  elimasn 2617  imaiun 2650  dmco2 2673  tz9.12lem1 3503

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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