Theorem bj-fununsn1 37297

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 6161	. . . 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 4590	. . . 4 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
6	4, 5	nsyl 140	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵})
7	3, 6	ssneldd 3932	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ dom {⟨𝐵, 𝐶⟩})
8	1, 7	bj-funun 37296	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2111   ∪ cun 3895   ⊆ wss 3897  {csn 4573  ⟨cop 4579  dom cdm 5614  ‘cfv 6481

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fv 6489

This theorem is referenced by:  bj-fvsnun1  37299  bj-fvmptunsn2  37302