Theorem fvun2d 6911

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 511	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵))
6	1, 2, 5	3jca 1128	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)))
7		fvun2 6909	. 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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110   ∪ cun 3898   ∩ cin 3899  ∅c0 4281   Fn wfn 6472  ‘cfv 6477

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-fv 6485

This theorem is referenced by:  noetainflem4  27672  ofun  42248  tfsconcatfv2  43352