Theorem bj-funun 37396

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 6012	. . . . 5 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 ∪ 𝐻) “ {𝐴}))
3		imaundir 6106	. . . . 5 ⊢ ((𝐺 ∪ 𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴}))
4	2, 3	eqtrdi 2785	. . . 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 6060	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝐻 “ {𝐴}) = ∅)
9		uneq2 4112	. . . . 5 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅))
10		un0 4344	. . . . 5 ⊢ ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴})
11	9, 10	eqtrdi 2785	. . . 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 2769	. 2 ⊢ (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴}))
14		bj-imafv 37395	. 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 2113   ∪ cun 3897  ∅c0 4283  {csn 4578  dom cdm 5622   “ cima 5625  ‘cfv 6490

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-xp 5628  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fv 6498

This theorem is referenced by:  bj-fununsn1  37397  bj-fununsn2  37398