Theorem bj-fununsn1 36655

Description: Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at a point not equal to the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-fununsn.un	⊢ (𝜑 → 𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))
bj-fununsn1.neq	⊢ (𝜑 → ¬ 𝐴 = 𝐵)

Assertion

Ref	Expression
bj-fununsn1	⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Proof of Theorem bj-fununsn1

Step	Hyp	Ref	Expression
1		bj-fununsn.un	. 2 ⊢ (𝜑 → 𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))
2		dmsnopss 6212	. . . 4 ⊢ dom {⟨𝐵, 𝐶⟩} ⊆ {𝐵}
3	2	a1i 11	. . 3 ⊢ (𝜑 → dom {⟨𝐵, 𝐶⟩} ⊆ {𝐵})
4		bj-fununsn1.neq	. . . 4 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
5		elsni 4641	. . . 4 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
6	4, 5	nsyl 140	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵})
7	3, 6	ssneldd 3981	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ dom {⟨𝐵, 𝐶⟩})
8	1, 7	bj-funun 36654	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534   ∈ wcel 2099   ∪ cun 3942   ⊆ wss 3944  {csn 4624  ⟨cop 4630  dom cdm 5672  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fv 6550

This theorem is referenced by:  bj-fvsnun1  36657  bj-fvmptunsn2  36660