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Theorem fvun2 6432
Description: The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
Assertion
Ref Expression
fvun2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))

Proof of Theorem fvun2
StepHypRef Expression
1 uncom 3900 . . 3 (𝐹𝐺) = (𝐺𝐹)
21fveq1i 6353 . 2 ((𝐹𝐺)‘𝑋) = ((𝐺𝐹)‘𝑋)
3 incom 3948 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
43eqeq1i 2765 . . . . 5 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
54anbi1i 733 . . . 4 (((𝐴𝐵) = ∅ ∧ 𝑋𝐵) ↔ ((𝐵𝐴) = ∅ ∧ 𝑋𝐵))
6 fvun1 6431 . . . 4 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ ((𝐵𝐴) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
75, 6syl3an3b 1512 . . 3 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
873com12 1118 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
92, 8syl5eq 2806 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  cun 3713  cin 3714  c0 4058   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  fveqf1o  6720  xpsc1  16423  ptunhmeo  21813  axlowdimlem9  26029  axlowdimlem12  26032  axlowdimlem17  26037  vtxdun  26587  isoun  29788  resf1o  29814  sseqfv2  30765  actfunsnrndisj  30992  reprsuc  31002  breprexplema  31017  cvmliftlem4  31577  noextenddif  32127  noextendlt  32128  noextendgt  32129  noetalem3  32171  fullfunfv  32360  finixpnum  33707  poimirlem1  33723  poimirlem2  33724  poimirlem3  33725  poimirlem4  33726  poimirlem6  33728  poimirlem7  33729  poimirlem11  33733  poimirlem12  33734  poimirlem16  33738  poimirlem19  33741  poimirlem20  33742  poimirlem23  33745  poimirlem28  33750
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