Theorem fvun2d 6998

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	⊢ (𝜑 → 𝐹 Fn 𝐴)
fvun2d.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
fvun2d.3	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
fvun2d.4	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
fvun2d	⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))

Proof of Theorem fvun2d

Step	Hyp	Ref	Expression
1		fvun2d.1	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fvun2d.2	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
3		fvun2d.3	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
4		fvun2d.4	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5	3, 4	jca 510	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵))
6	1, 2, 5	3jca 1125	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)))
7		fvun2 6996	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))
8	6, 7	syl 17	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ∪ cun 3945   ∩ cin 3946  ∅c0 4325   Fn wfn 6551  ‘cfv 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-fv 6564

This theorem is referenced by:  noetainflem4  27773  metakunt20  41912  metakunt22  41914  ofun  41962  tfsconcatfv2  43024