Theorem bj-fununsn1 37938

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 6220	. . . 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 4611	. . . 4 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
6	4, 5	nsyl 141	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵})
7	3, 6	ssneldd 3943	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ dom {⟨𝐵, 𝐶⟩})
8	1, 7	bj-funun 37937	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146   ∪ cun 3906   ⊆ wss 3908  {csn 4594  ⟨cop 4600  dom cdm 5666  ‘cfv 6543

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 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fv 6551

This theorem is used by:  bj-fvsnun1  37940  bj-fvmptunsn2  37943