Theorem fvun1d 6969

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	⊢ (𝜑 → 𝐹 Fn 𝐴)
fvun1d.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
fvun1d.3	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
fvun1d.4	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
fvun1d	⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))

Proof of Theorem fvun1d

Step	Hyp	Ref	Expression
1		fvun1d.1	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fvun1d.2	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
3		fvun1d.3	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
4		fvun1d.4	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
5	3, 4	jca 512	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴))
6	1, 2, 5	3jca 1128	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)))
7		fvun1 6967	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
8	6, 7	syl 17	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∪ cun 3941   ∩ cin 3942  ∅c0 4317   Fn wfn 6526  ‘cfv 6531

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5141  df-opab 5203  df-id 5566  df-xp 5674  df-rel 5675  df-cnv 5676  df-co 5677  df-dm 5678  df-rn 5679  df-res 5680  df-ima 5681  df-iota 6483  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is referenced by:  hashf1lem1  14396  hashf1lem1OLD  14397  elrspunidl  32393  metakunt21  40796  metakunt22  40797  ofun  40856  tfsconcatfv1  41848