Theorem bj-funun 36121

Description: Value of a function expressed as a union of two functions at a point not in the domain of one of them. (Contributed by BJ, 18-Mar-2023.)

Hypotheses

Ref	Expression
bj-funun.un	⊢ (𝜑 → 𝐹 = (𝐺 ∪ 𝐻))
bj-funun.neldm	⊢ (𝜑 → ¬ 𝐴 ∈ dom 𝐻)

Assertion

Ref	Expression
bj-funun	⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Proof of Theorem bj-funun

Step	Hyp	Ref	Expression
1		bj-funun.un	. . . 4 ⊢ (𝜑 → 𝐹 = (𝐺 ∪ 𝐻))
2		imaeq1 6052	. . . . 5 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 ∪ 𝐻) “ {𝐴}))
3		imaundir 6147	. . . . 5 ⊢ ((𝐺 ∪ 𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴}))
4	2, 3	eqtrdi 2788	. . . 4 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})))
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})))
6		bj-funun.neldm	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 ∈ dom 𝐻)
7		ndmima 6099	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝐻 “ {𝐴}) = ∅)
9		uneq2 4156	. . . . 5 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅))
10		un0 4389	. . . . 5 ⊢ ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴})
11	9, 10	eqtrdi 2788	. . . 4 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
12	8, 11	syl 17	. . 3 ⊢ (𝜑 → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
13	5, 12	eqtrd 2772	. 2 ⊢ (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴}))
14		bj-imafv 36120	. 2 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹‘𝐴) = (𝐺‘𝐴))
15	13, 14	syl 17	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2106   ∪ cun 3945  ∅c0 4321  {csn 4627  dom cdm 5675   “ cima 5678  ‘cfv 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fv 6548

This theorem is referenced by:  bj-fununsn1  36122  bj-fununsn2  36123