Theorem fvun1d 7002

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 511	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴))
6	1, 2, 5	3jca 1129	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)))
7		fvun1 7000	. 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 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ∪ cun 3949   ∩ cin 3950  ∅c0 4333   Fn wfn 6556  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is referenced by:  hashf1lem1  14494  elrspunidl  33456  metakunt21  42226  metakunt22  42227  ofun  42277  tfsconcatfv1  43352