Theorem bj-fununsn2 35352

Description: Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at 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-fununsn2.neldm	⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
bj-fununsn2.ex1	⊢ (𝜑 → 𝐵 ∈ 𝑉)
bj-fununsn2.ex2	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

Ref	Expression
bj-fununsn2	⊢ (𝜑 → (𝐹‘𝐵) = 𝐶)

Proof of Theorem bj-fununsn2

Step	Hyp	Ref	Expression
1		bj-fununsn.un	. . . 4 ⊢ (𝜑 → 𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))
2		uncom 4083	. . . 4 ⊢ (𝐺 ∪ {⟨𝐵, 𝐶⟩}) = ({⟨𝐵, 𝐶⟩} ∪ 𝐺)
3	1, 2	eqtrdi 2795	. . 3 ⊢ (𝜑 → 𝐹 = ({⟨𝐵, 𝐶⟩} ∪ 𝐺))
4		bj-fununsn2.neldm	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
5	3, 4	bj-funun 35350	. 2 ⊢ (𝜑 → (𝐹‘𝐵) = ({⟨𝐵, 𝐶⟩}‘𝐵))
6		bj-fununsn2.ex1	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		bj-fununsn2.ex2	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
8		fvsng 7034	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
9	6, 7, 8	syl2anc 583	. 2 ⊢ (𝜑 → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
10	5, 9	eqtrd 2778	1 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2108   ∪ cun 3881  {csn 4558  ⟨cop 4564  dom cdm 5580  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by:  bj-fvsnun2  35354  bj-fvmptunsn1  35355