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Theorem fvun2 6752
 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 4133 . . 3 (𝐹𝐺) = (𝐺𝐹)
21fveq1i 6668 . 2 ((𝐹𝐺)‘𝑋) = ((𝐺𝐹)‘𝑋)
3 incom 4182 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
43eqeq1i 2831 . . . . 5 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
54anbi1i 623 . . . 4 (((𝐴𝐵) = ∅ ∧ 𝑋𝐵) ↔ ((𝐵𝐴) = ∅ ∧ 𝑋𝐵))
6 fvun1 6751 . . . 4 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ ((𝐵𝐴) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
75, 6syl3an3b 1399 . . 3 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
873com12 1117 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
92, 8syl5eq 2873 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ∪ cun 3938   ∩ cin 3939  ∅c0 4295   Fn wfn 6347  ‘cfv 6352 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360 This theorem is referenced by:  fveqf1o  7052  ptunhmeo  22332  axlowdimlem9  26650  axlowdimlem12  26653  axlowdimlem17  26658  vtxdun  27177  isoun  30350  resf1o  30379  cycpmfvlem  30668  lbsdiflsp0  30908  sseqfv2  31538  actfunsnrndisj  31762  reprsuc  31772  breprexplema  31787  cvmliftlem4  32419  frrlem12  33018  noextenddif  33059  noextendlt  33060  noextendgt  33061  noetalem3  33103  fullfunfv  33292  finixpnum  34744  poimirlem1  34760  poimirlem2  34761  poimirlem3  34762  poimirlem4  34763  poimirlem6  34765  poimirlem7  34766  poimirlem11  34770  poimirlem12  34771  poimirlem16  34775  poimirlem19  34778  poimirlem20  34779  poimirlem23  34782  poimirlem28  34787
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