Theorem fvun2d 6965

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 520	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵))
6	1, 2, 5	3jca 1144	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)))
7		fvun2 6963	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))
8	6, 7	syl 18	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ∪ cun 3905   ∩ cin 3906  ∅c0 4288   Fn wfn 6520  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533

This theorem is referenced by:  noetainflem4  27862  ofun  42866  tfsconcatfv2  43929