Theorem bj-fununsn2 36135

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 4154	. . . 4 ⊢ (𝐺 ∪ {⟨𝐵, 𝐶⟩}) = ({⟨𝐵, 𝐶⟩} ∪ 𝐺)
3	1, 2	eqtrdi 2789	. . 3 ⊢ (𝜑 → 𝐹 = ({⟨𝐵, 𝐶⟩} ∪ 𝐺))
4		bj-fununsn2.neldm	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
5	3, 4	bj-funun 36133	. 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 7178	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
9	6, 7, 8	syl2anc 585	. 2 ⊢ (𝜑 → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
10	5, 9	eqtrd 2773	1 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2107   ∪ cun 3947  {csn 4629  ⟨cop 4635  dom cdm 5677  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552

This theorem is referenced by:  bj-fvsnun2  36137  bj-fvmptunsn1  36138