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Theorem fvun2 5165
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 A 𝐺 Fn B ((AB) = ∅ 𝑋 B)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))

Proof of Theorem fvun2
StepHypRef Expression
1 uncom 3064 . . 3 (𝐹𝐺) = (𝐺𝐹)
21fveq1i 5104 . 2 ((𝐹𝐺)‘𝑋) = ((𝐺𝐹)‘𝑋)
3 incom 3106 . . . . . 6 (AB) = (BA)
43eqeq1i 2029 . . . . 5 ((AB) = ∅ ↔ (BA) = ∅)
54anbi1i 434 . . . 4 (((AB) = ∅ 𝑋 B) ↔ ((BA) = ∅ 𝑋 B))
6 fvun1 5164 . . . 4 ((𝐺 Fn B 𝐹 Fn A ((BA) = ∅ 𝑋 B)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
75, 6syl3an3b 1159 . . 3 ((𝐺 Fn B 𝐹 Fn A ((AB) = ∅ 𝑋 B)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
873com12 1094 . 2 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 B)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
92, 8syl5eq 2066 1 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 B)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873   = wceq 1228   wcel 1374  cun 2892  cin 2893  c0 3201   Fn wfn 4824  cfv 4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by: (None)
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