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Theorem funimaexg 5875
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 5368 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
21eleq1d 2671 . . . 4 (𝑤 = 𝐵 → ((𝐴𝑤) ∈ V ↔ (𝐴𝐵) ∈ V))
32imbi2d 328 . . 3 (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴𝐵) ∈ V)))
4 dffun5 5803 . . . . 5 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
54simprbi 478 . . . 4 (Fun 𝐴 → ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
6 nfv 1829 . . . . . 6 𝑧𝑥, 𝑦⟩ ∈ 𝐴
76axrep4 4697 . . . . 5 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
8 isset 3179 . . . . . 6 ((𝐴𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴𝑤))
9 dfima3 5375 . . . . . . . . 9 (𝐴𝑤) = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
109eqeq2i 2621 . . . . . . . 8 (𝑧 = (𝐴𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
11 abeq2 2718 . . . . . . . 8 (𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1210, 11bitri 262 . . . . . . 7 (𝑧 = (𝐴𝑤) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1312exbii 1763 . . . . . 6 (∃𝑧 𝑧 = (𝐴𝑤) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
148, 13bitri 262 . . . . 5 ((𝐴𝑤) ∈ V ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
157, 14sylibr 222 . . . 4 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → (𝐴𝑤) ∈ V)
165, 15syl 17 . . 3 (Fun 𝐴 → (𝐴𝑤) ∈ V)
173, 16vtoclg 3238 . 2 (𝐵𝐶 → (Fun 𝐴 → (𝐴𝐵) ∈ V))
1817impcom 444 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976  {cab 2595  Vcvv 3172  cop 4130  cima 5031  Rel wrel 5033  Fun wfun 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-fun 5792
This theorem is referenced by:  funimaex  5876  resfunexg  6362  resfunexgALT  6999  fnexALT  7002  wdomimag  8352  carduniima  8779  dfac12lem2  8826  ttukeylem3  9193  nnexALT  10869  seqex  12620  fbasrn  21440  elfm3  21506  nobndlem1  30897
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