MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funimaexg Structured version   Visualization version   GIF version

Theorem funimaexg 6653
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 2138, ax-12 2174. (Revised by SN, 19-Dec-2024.)
Assertion
Ref Expression
funimaexg ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Proof of Theorem funimaexg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun6 6575 . . . 4 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
21simprbi 496 . . 3 (Fun 𝐴 → ∀𝑥∃*𝑦 𝑥𝐴𝑦)
3 dfima2 6081 . . . 4 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
4 axrep6g 5295 . . . 4 ((𝐵𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦} ∈ V)
53, 4eqeltrid 2842 . . 3 ((𝐵𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → (𝐴𝐵) ∈ V)
62, 5sylan2 593 . 2 ((𝐵𝐶 ∧ Fun 𝐴) → (𝐴𝐵) ∈ V)
76ancoms 458 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1534  wcel 2105  ∃*wmo 2535  {cab 2711  wrex 3067  Vcvv 3477   class class class wbr 5147  cima 5691  Rel wrel 5693  Fun wfun 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-fun 6564
This theorem is referenced by:  funimaex  6655  resfunexg  7234  resfunexgALT  7970  fnexALT  7973  naddcllem  8712  naddunif  8729  wdomimag  9624  carduniima  10133  dfac12lem2  10182  ttukeylem3  10548  nnexALT  12265  seqex  14040  fbasrn  23907  elfm3  23973  bdayimaon  27752  nosupno  27762  noinfno  27777  noeta2  27843  etasslt2  27873  scutbdaybnd2lim  27876  madeval  27905  oldval  27907  negsunif  28101  fnimafnex  43429  fundcmpsurinjlem3  47324  fundcmpsurbijinjpreimafv  47331  fundcmpsurbijinj  47334  fundcmpsurinjALT  47336  uspgrimprop  47810  grimuhgr  47815
  Copyright terms: Public domain W3C validator