Theorem funimaexg 6573

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 2146, ax-12 2182. (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 6497	. . . 4 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2	1	simprbi 496	. . 3 ⊢ (Fun 𝐴 → ∀𝑥∃*𝑦 𝑥𝐴𝑦)
3		dfima2 6015	. . . 4 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦}
4		axrep6g 5230	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} ∈ V)
5	3, 4	eqeltrid 2837	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → (𝐴 “ 𝐵) ∈ V)
6	2, 5	sylan2 593	. 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 1539   ∈ wcel 2113  ∃*wmo 2535  {cab 2711  ∃wrex 3057  Vcvv 3437   class class class wbr 5093   “ cima 5622  Rel wrel 5624  Fun wfun 6480

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6488

This theorem is referenced by:  funimaex  6574  resfunexg  7155  resfunexgALT  7886  fnexALT  7889  naddcllem  8597  naddunif  8614  wdomimag  9480  carduniima  9994  dfac12lem2  10043  ttukeylem3  10409  nnexALT  12134  seqex  13912  fbasrn  23800  elfm3  23866  bdayimaon  27633  nosupno  27643  noinfno  27658  noeta2  27725  etasslt2  27756  scutbdaybnd2lim  27759  madeval  27794  oldval  27796  negsunif  27998  bdayon  28210  fnimafnex  43557  fundcmpsurinjlem3  47524  fundcmpsurbijinjpreimafv  47531  fundcmpsurbijinj  47534  fundcmpsurinjALT  47536  grimuhgr  48011