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Theorem fvun2d 7002
Description: The value of a union when the argument is in the second domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
Hypotheses
Ref Expression
fvun2d.1 (𝜑𝐹 Fn 𝐴)
fvun2d.2 (𝜑𝐺 Fn 𝐵)
fvun2d.3 (𝜑 → (𝐴𝐵) = ∅)
fvun2d.4 (𝜑𝑋𝐵)
Assertion
Ref Expression
fvun2d (𝜑 → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))

Proof of Theorem fvun2d
StepHypRef Expression
1 fvun2d.1 . . 3 (𝜑𝐹 Fn 𝐴)
2 fvun2d.2 . . 3 (𝜑𝐺 Fn 𝐵)
3 fvun2d.3 . . . 4 (𝜑 → (𝐴𝐵) = ∅)
4 fvun2d.4 . . . 4 (𝜑𝑋𝐵)
53, 4jca 511 . . 3 (𝜑 → ((𝐴𝐵) = ∅ ∧ 𝑋𝐵))
61, 2, 53jca 1127 . 2 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)))
7 fvun2 7000 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
86, 7syl 17 1 (𝜑 → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  cun 3960  cin 3961  c0 4338   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  noetainflem4  27799  metakunt20  42205  metakunt22  42207  ofun  42255  tfsconcatfv2  43329
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