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Theorem funimaexg 5935
Description: Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
Assertion
Ref Expression
funimaexg ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Proof of Theorem funimaexg
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imaeq2 5425 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
21eleq1d 2688 . . . 4 (𝑤 = 𝐵 → ((𝐴𝑤) ∈ V ↔ (𝐴𝐵) ∈ V))
32imbi2d 330 . . 3 (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴𝐵) ∈ V)))
4 dffun5 5863 . . . . 5 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
54simprbi 480 . . . 4 (Fun 𝐴 → ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
6 nfv 1845 . . . . . 6 𝑧𝑥, 𝑦⟩ ∈ 𝐴
76axrep4 4740 . . . . 5 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
8 isset 3198 . . . . . 6 ((𝐴𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴𝑤))
9 dfima3 5432 . . . . . . . . 9 (𝐴𝑤) = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
109eqeq2i 2638 . . . . . . . 8 (𝑧 = (𝐴𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
11 abeq2 2735 . . . . . . . 8 (𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1210, 11bitri 264 . . . . . . 7 (𝑧 = (𝐴𝑤) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1312exbii 1772 . . . . . 6 (∃𝑧 𝑧 = (𝐴𝑤) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
148, 13bitri 264 . . . . 5 ((𝐴𝑤) ∈ V ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
157, 14sylibr 224 . . . 4 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → (𝐴𝑤) ∈ V)
165, 15syl 17 . . 3 (Fun 𝐴 → (𝐴𝑤) ∈ V)
173, 16vtoclg 3257 . 2 (𝐵𝐶 → (Fun 𝐴 → (𝐴𝐵) ∈ V))
1817impcom 446 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1992  {cab 2612  Vcvv 3191  cop 4159  cima 5082  Rel wrel 5084  Fun wfun 5844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-fun 5852
This theorem is referenced by:  funimaex  5936  resfunexg  6434  resfunexgALT  7079  fnexALT  7082  wdomimag  8437  carduniima  8864  dfac12lem2  8911  ttukeylem3  9278  nnexALT  10967  seqex  12740  fbasrn  21593  elfm3  21659  nobndlem1  31536
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