Theorem funimaexg 6579

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 2152, ax-12 2189. (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 6503	. . . 4 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2	1	simprbi 498	. . 3 ⊢ (Fun 𝐴 → ∀𝑥∃*𝑦 𝑥𝐴𝑦)
3		dfima2 6021	. . . 4 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦}
4		axrep6g 5219	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} ∈ V)
5	3, 4	eqeltrid 2844	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → (𝐴 “ 𝐵) ∈ V)
6	2, 5	sylan2 599	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Fun 𝐴) → (𝐴 “ 𝐵) ∈ V)
7	6	ancoms 459	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   ∈ wcel 2119  ∃*wmo 2541  {cab 2718  ∃wrex 3064  Vcvv 3432   class class class wbr 5079   “ cima 5628  Rel wrel 5630  Fun wfun 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494

This theorem is referenced by:  funimaex  6580  resfunexg  7166  resfunexgALT  7897  fnexALT  7900  naddcllem  8609  naddunif  8626  wdomimag  9499  carduniima  10016  dfac12lem2  10065  ttukeylem3  10431  nnexALT  12174  seqex  13963  fbasrn  23874  elfm3  23940  bdayimaon  27682  nosupno  27692  noinfno  27707  noeta2  27778  etaslts2  27811  cutbdaybnd2lim  27814  madeval  27849  oldval  27851  negsunif  28072  bdayons  28293  n0sexg  28333  fnimafnex  43891  fundcmpsurinjlem3  47882  fundcmpsurbijinjpreimafv  47889  fundcmpsurbijinj  47892  fundcmpsurinjALT  47894  grimuhgr  48385