Theorem funimaexg 6440

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 5925	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 “ 𝑤) = (𝐴 “ 𝐵))
2	1	eleq1d 2897	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝐴 “ 𝑤) ∈ V ↔ (𝐴 “ 𝐵) ∈ V))
3	2	imbi2d 343	. . 3 ⊢ (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴 “ 𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)))
4		dffun5 6368	. . . 4 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)))
5		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩ ∈ 𝐴
6	5	axrep4 5195	. . . . 5 ⊢ (∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
7		isset 3506	. . . . . 6 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴 “ 𝑤))
8		dfima3 5932	. . . . . . . . 9 ⊢ (𝐴 “ 𝑤) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
9	8	eqeq2i 2834	. . . . . . . 8 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
10		abeq2 2945	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
11	9, 10	bitri 277	. . . . . . 7 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
12	11	exbii 1848	. . . . . 6 ⊢ (∃𝑧 𝑧 = (𝐴 “ 𝑤) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
13	7, 12	bitri 277	. . . . 5 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
14	6, 13	sylibr 236	. . . 4 ⊢ (∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧) → (𝐴 “ 𝑤) ∈ V)
15	4, 14	simplbiim 507	. . 3 ⊢ (Fun 𝐴 → (𝐴 “ 𝑤) ∈ V)
16	3, 15	vtoclg 3567	. 2 ⊢ (𝐵 ∈ 𝐶 → (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V))
17	16	impcom 410	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799  Vcvv 3494  ⟨cop 4573   “ cima 5558  Rel wrel 5560  Fun wfun 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-fun 6357

This theorem is referenced by:  funimaex  6441  resfunexg  6978  resfunexgALT  7649  fnexALT  7652  wdomimag  9051  carduniima  9522  dfac12lem2  9570  ttukeylem3  9933  nnexALT  11640  seqex  13372  fbasrn  22492  elfm3  22558  bdayimaon  33197  nosupno  33203  madeval  33289  fundcmpsurinjlem3  43580  fundcmpsurbijinjpreimafv  43587  fundcmpsurbijinj  43590  fundcmpsurinjALT  43592