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Theorem funimaexg 6208
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 5703 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
21eleq1d 2891 . . . 4 (𝑤 = 𝐵 → ((𝐴𝑤) ∈ V ↔ (𝐴𝐵) ∈ V))
32imbi2d 332 . . 3 (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴𝐵) ∈ V)))
4 dffun5 6136 . . . . 5 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
54simprbi 492 . . . 4 (Fun 𝐴 → ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
6 nfv 2013 . . . . . 6 𝑧𝑥, 𝑦⟩ ∈ 𝐴
76axrep4 4999 . . . . 5 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
8 isset 3424 . . . . . 6 ((𝐴𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴𝑤))
9 dfima3 5710 . . . . . . . . 9 (𝐴𝑤) = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
109eqeq2i 2837 . . . . . . . 8 (𝑧 = (𝐴𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
11 abeq2 2937 . . . . . . . 8 (𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1210, 11bitri 267 . . . . . . 7 (𝑧 = (𝐴𝑤) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1312exbii 1947 . . . . . 6 (∃𝑧 𝑧 = (𝐴𝑤) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
148, 13bitri 267 . . . . 5 ((𝐴𝑤) ∈ V ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
157, 14sylibr 226 . . . 4 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → (𝐴𝑤) ∈ V)
165, 15syl 17 . . 3 (Fun 𝐴 → (𝐴𝑤) ∈ V)
173, 16vtoclg 3482 . 2 (𝐵𝐶 → (Fun 𝐴 → (𝐴𝐵) ∈ V))
1817impcom 398 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1654   = wceq 1656  wex 1878  wcel 2164  {cab 2811  Vcvv 3414  cop 4403  cima 5345  Rel wrel 5347  Fun wfun 6117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-fun 6125
This theorem is referenced by:  funimaex  6209  resfunexg  6735  resfunexgALT  7391  fnexALT  7394  wdomimag  8761  carduniima  9232  dfac12lem2  9281  ttukeylem3  9648  nnexALT  11352  seqex  13097  fbasrn  22058  elfm3  22124  bdayimaon  32371  nosupno  32377  madeval  32463
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