Theorem funimaexg 6421

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 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Proof of Theorem funimaexg

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaeq2 5897	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 “ 𝑤) = (𝐴 “ 𝐵))
2	1	eleq1d 2836	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝐴 “ 𝑤) ∈ V ↔ (𝐴 “ 𝐵) ∈ V))
3	2	imbi2d 344	. . 3 ⊢ (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴 “ 𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)))
4		dffun5 6348	. . . 4 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)))
5		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩ ∈ 𝐴
6	5	axrep4 5161	. . . . 5 ⊢ (∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
7		isset 3422	. . . . . 6 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴 “ 𝑤))
8		dfima3 5904	. . . . . . . . 9 ⊢ (𝐴 “ 𝑤) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
9	8	eqeq2i 2771	. . . . . . . 8 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
10		abeq2 2884	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
11	9, 10	bitri 278	. . . . . . 7 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
12	11	exbii 1849	. . . . . 6 ⊢ (∃𝑧 𝑧 = (𝐴 “ 𝑤) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
13	7, 12	bitri 278	. . . . 5 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
14	6, 13	sylibr 237	. . . 4 ⊢ (∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧) → (𝐴 “ 𝑤) ∈ V)
15	4, 14	simplbiim 508	. . 3 ⊢ (Fun 𝐴 → (𝐴 “ 𝑤) ∈ V)
16	3, 15	vtoclg 3485	. 2 ⊢ (𝐵 ∈ 𝐶 → (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V))
17	16	impcom 411	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  Vcvv 3409  ⟨cop 4528   “ cima 5527  Rel wrel 5529  Fun wfun 6329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-fun 6337

This theorem is referenced by:  funimaex  6422  resfunexg  6969  resfunexgALT  7653  fnexALT  7656  wdomimag  9084  carduniima  9556  dfac12lem2  9604  ttukeylem3  9971  nnexALT  11676  seqex  13420  fbasrn  22584  elfm3  22650  bdayimaon  33461  nosupno  33471  noinfno  33486  noeta2  33544  etasslt2  33569  scutbdaybnd2lim  33572  madeval  33596  oldval  33598  fundcmpsurinjlem3  44285  fundcmpsurbijinjpreimafv  44292  fundcmpsurbijinj  44295  fundcmpsurinjALT  44297