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Theorem fvun1d 6735
 Description: The value of a union when the argument is in the first domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
Hypotheses
Ref Expression
fvun1d.1 (𝜑𝐹 Fn 𝐴)
fvun1d.2 (𝜑𝐺 Fn 𝐵)
fvun1d.3 (𝜑 → (𝐴𝐵) = ∅)
fvun1d.4 (𝜑𝑋𝐴)
Assertion
Ref Expression
fvun1d (𝜑 → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))

Proof of Theorem fvun1d
StepHypRef Expression
1 fvun1d.1 . . 3 (𝜑𝐹 Fn 𝐴)
2 fvun1d.2 . . 3 (𝜑𝐺 Fn 𝐵)
3 fvun1d.3 . . . 4 (𝜑 → (𝐴𝐵) = ∅)
4 fvun1d.4 . . . 4 (𝜑𝑋𝐴)
53, 4jca 515 . . 3 (𝜑 → ((𝐴𝐵) = ∅ ∧ 𝑋𝐴))
61, 2, 53jca 1125 . 2 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)))
7 fvun1 6733 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
86, 7syl 17 1 (𝜑 → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ∪ cun 3882   ∩ cin 3883  ∅c0 4246   Fn wfn 6323  ‘cfv 6328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336 This theorem is referenced by:  elrspunidl  31018  metakunt21  39367  metakunt22  39368  ofun  39409
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