Theorem fvun1d 6924

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 1128	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)))
7		fvun1 6922	. 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 1086   = wceq 1541   ∈ wcel 2113   ∪ cun 3897   ∩ cin 3898  ∅c0 4284   Fn wfn 6484  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497

This theorem is referenced by:  hashf1lem1  14372  elrspunidl  33404  ofun  42344  tfsconcatfv1  43446