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Theorem fvun2 6495
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 3955 . . 3 (𝐹𝐺) = (𝐺𝐹)
21fveq1i 6412 . 2 ((𝐹𝐺)‘𝑋) = ((𝐺𝐹)‘𝑋)
3 incom 4003 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
43eqeq1i 2804 . . . . 5 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
54anbi1i 618 . . . 4 (((𝐴𝐵) = ∅ ∧ 𝑋𝐵) ↔ ((𝐵𝐴) = ∅ ∧ 𝑋𝐵))
6 fvun1 6494 . . . 4 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ ((𝐵𝐴) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
75, 6syl3an3b 1525 . . 3 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
873com12 1154 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
92, 8syl5eq 2845 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  cun 3767  cin 3768  c0 4115   Fn wfn 6096  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109
This theorem is referenced by:  fveqf1o  6785  xpsc1  16536  ptunhmeo  21940  axlowdimlem9  26187  axlowdimlem12  26190  axlowdimlem17  26195  vtxdun  26731  isoun  29997  resf1o  30023  sseqfv2  30973  actfunsnrndisj  31203  reprsuc  31213  breprexplema  31228  cvmliftlem4  31787  noextenddif  32334  noextendlt  32335  noextendgt  32336  noetalem3  32378  fullfunfv  32567  finixpnum  33883  poimirlem1  33899  poimirlem2  33900  poimirlem3  33901  poimirlem4  33902  poimirlem6  33904  poimirlem7  33905  poimirlem11  33909  poimirlem12  33910  poimirlem16  33914  poimirlem19  33917  poimirlem20  33918  poimirlem23  33921  poimirlem28  33926
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