Theorem bj-funun 35350

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 5953	. . . . 5 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 ∪ 𝐻) “ {𝐴}))
3		imaundir 6043	. . . . 5 ⊢ ((𝐺 ∪ 𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴}))
4	2, 3	eqtrdi 2795	. . . 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 6000	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝐻 “ {𝐴}) = ∅)
9		uneq2 4087	. . . . 5 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅))
10		un0 4321	. . . . 5 ⊢ ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴})
11	9, 10	eqtrdi 2795	. . . 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 2778	. 2 ⊢ (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴}))
14		bj-imafv 35349	. 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 1539   ∈ wcel 2108   ∪ cun 3881  ∅c0 4253  {csn 4558  dom cdm 5580   “ cima 5583  ‘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-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-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fv 6426

This theorem is referenced by:  bj-fununsn1  35351  bj-fununsn2  35352