Theorem fvun2d 6950

Description: The value of a union when the argument is in the second domain, a deduction version. (Contributed by metakunt, 28-May-2024.)

Hypotheses

Ref	Expression
fvun2d.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fvun2d.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
fvun2d.3	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
fvun2d.4	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
fvun2d	⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))

Proof of Theorem fvun2d

Step	Hyp	Ref	Expression
1		fvun2d.1	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fvun2d.2	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
3		fvun2d.3	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
4		fvun2d.4	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5	3, 4	jca 518	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵))
6	1, 2, 5	3jca 1137	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)))
7		fvun2 6948	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))
8	6, 7	syl 17	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ∪ cun 3897   ∩ cin 3898  ∅c0 4280   Fn wfn 6505  ‘cfv 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-fv 6518

This theorem is referenced by:  noetainflem4  27774  ofun  42802  tfsconcatfv2  43865