Theorem funimaexg 6585

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.) Shorten proof and avoid ax-10 2147, ax-12 2185. (Revised by SN, 19-Dec-2024.)

Assertion

Ref	Expression
funimaexg	⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Proof of Theorem funimaexg

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

Step	Hyp	Ref	Expression
1		dffun6 6509	. . . 4 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2	1	simprbi 497	. . 3 ⊢ (Fun 𝐴 → ∀𝑥∃*𝑦 𝑥𝐴𝑦)
3		dfima2 6027	. . . 4 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦}
4		axrep6g 5225	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} ∈ V)
5	3, 4	eqeltrid 2840	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → (𝐴 “ 𝐵) ∈ V)
6	2, 5	sylan2 594	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Fun 𝐴) → (𝐴 “ 𝐵) ∈ V)
7	6	ancoms 458	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   ∈ wcel 2114  ∃*wmo 2537  {cab 2714  ∃wrex 3061  Vcvv 3429   class class class wbr 5085   “ cima 5634  Rel wrel 5636  Fun wfun 6492

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6500

This theorem is referenced by:  funimaex  6586  resfunexg  7170  resfunexgALT  7901  fnexALT  7904  naddcllem  8612  naddunif  8629  wdomimag  9502  carduniima  10018  dfac12lem2  10067  ttukeylem3  10433  nnexALT  12176  seqex  13965  fbasrn  23849  elfm3  23915  bdayimaon  27657  nosupno  27667  noinfno  27682  noeta2  27753  etaslts2  27786  cutbdaybnd2lim  27789  madeval  27824  oldval  27826  negsunif  28047  bdayons  28268  n0sexg  28308  fnimafnex  43867  fundcmpsurinjlem3  47860  fundcmpsurbijinjpreimafv  47867  fundcmpsurbijinj  47870  fundcmpsurinjALT  47872  grimuhgr  48363