Theorem fvun1d 6843

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 1126	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)))
7		fvun1 6841	. 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 1085   = wceq 1539   ∈ wcel 2108   ∪ cun 3881   ∩ cin 3882  ∅c0 4253   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  hashf1lem1  14096  hashf1lem1OLD  14097  elrspunidl  31508  metakunt21  40073  metakunt22  40074  ofun  40137