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Theorem fvun2 5456
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 3190 . . 3 (𝐹𝐺) = (𝐺𝐹)
21fveq1i 5390 . 2 ((𝐹𝐺)‘𝑋) = ((𝐺𝐹)‘𝑋)
3 incom 3238 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
43eqeq1i 2125 . . . . 5 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
54anbi1i 453 . . . 4 (((𝐴𝐵) = ∅ ∧ 𝑋𝐵) ↔ ((𝐵𝐴) = ∅ ∧ 𝑋𝐵))
6 fvun1 5455 . . . 4 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ ((𝐵𝐴) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
75, 6syl3an3b 1239 . . 3 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
873com12 1170 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐺𝐹)‘𝑋) = (𝐺𝑋))
92, 8syl5eq 2162 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947   = wceq 1316  wcel 1465  cun 3039  cin 3040  c0 3333   Fn wfn 5088  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  caseinr  6945
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