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Theorem funimaexg 5715
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 5214 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
21eleq1d 2567 . . . 4 (𝑤 = 𝐵 → ((𝐴𝑤) ∈ V ↔ (𝐴𝐵) ∈ V))
32imbi2d 325 . . 3 (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴𝐵) ∈ V)))
4 dffun5 5646 . . . . 5 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
54simprbi 473 . . . 4 (Fun 𝐴 → ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
6 nfv 1792 . . . . . 6 𝑧𝑥, 𝑦⟩ ∈ 𝐴
76axrep4 4552 . . . . 5 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
8 isset 3070 . . . . . 6 ((𝐴𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴𝑤))
9 dfima3 5221 . . . . . . . . 9 (𝐴𝑤) = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
109eqeq2i 2517 . . . . . . . 8 (𝑧 = (𝐴𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
11 abeq2 2614 . . . . . . . 8 (𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1210, 11bitri 259 . . . . . . 7 (𝑧 = (𝐴𝑤) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1312exbii 1749 . . . . . 6 (∃𝑧 𝑧 = (𝐴𝑤) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
148, 13bitri 259 . . . . 5 ((𝐴𝑤) ∈ V ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
157, 14sylibr 219 . . . 4 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → (𝐴𝑤) ∈ V)
165, 15syl 17 . . 3 (Fun 𝐴 → (𝐴𝑤) ∈ V)
173, 16vtoclg 3128 . 2 (𝐵𝐶 → (Fun 𝐴 → (𝐴𝐵) ∈ V))
1817impcom 439 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  wa 378  wal 1466   = wceq 1468  wex 1692  wcel 1937  {cab 2491  Vcvv 3066  cop 4001  cima 4883  Rel wrel 4885  Fun wfun 5627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-rep 4548  ax-sep 4558  ax-nul 4567  ax-pr 4680
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3068  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3758  df-if 3909  df-sn 3996  df-pr 3998  df-op 4002  df-br 4435  df-opab 4494  df-id 4795  df-xp 4886  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fun 5635
This theorem is referenced by:  funimaex  5716  resfunexg  6198  resfunexgALT  6833  fnexALT  6836  wdomimag  8185  carduniima  8612  dfac12lem2  8659  ttukeylem3  9026  nnexALT  10700  seqex  12329  fbasrn  21057  elfm3  21123  nobndlem1  30732
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