Theorem bj-funun 37953

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 6059	. . . . 5 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 ∪ 𝐻) “ {𝐴}))
3		imaundir 6150	. . . . 5 ⊢ ((𝐺 ∪ 𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴}))
4	2, 3	eqtrdi 2816	. . . 4 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})))
5	1, 4	syl 18	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})))
6		bj-funun.neldm	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 ∈ dom 𝐻)
7		ndmima 6107	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅)
8	6, 7	syl 18	. . . 4 ⊢ (𝜑 → (𝐻 “ {𝐴}) = ∅)
9		uneq2 4116	. . . . 5 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅))
10		un0 4351	. . . . 5 ⊢ ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴})
11	9, 10	eqtrdi 2816	. . . 4 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
12	8, 11	syl 18	. . 3 ⊢ (𝜑 → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
13	5, 12	eqtrd 2800	. 2 ⊢ (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴}))
14		bj-imafv 37952	. 2 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹‘𝐴) = (𝐺‘𝐴))
15	13, 14	syl 18	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146   ∪ cun 3904  ∅c0 4286  {csn 4591  dom cdm 5663   “ cima 5666  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fv 6548

This theorem is used by:  bj-fununsn1  37954  bj-fununsn2  37955