Theorem fvun1d 6984

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 1127	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)))
7		fvun1 6982	. 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 1540   ∈ wcel 2105   ∪ cun 3946   ∩ cin 3947  ∅c0 4322   Fn wfn 6538  ‘cfv 6543

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by:  hashf1lem1  14422  hashf1lem1OLD  14423  elrspunidl  32986  metakunt21  41472  metakunt22  41473  ofun  41525  tfsconcatfv1  42552