Theorem bj-fununsn2 36789

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 4146	. . . 4 ⊢ (𝐺 ∪ {⟨𝐵, 𝐶⟩}) = ({⟨𝐵, 𝐶⟩} ∪ 𝐺)
3	1, 2	eqtrdi 2781	. . 3 ⊢ (𝜑 → 𝐹 = ({⟨𝐵, 𝐶⟩} ∪ 𝐺))
4		bj-fununsn2.neldm	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
5	3, 4	bj-funun 36787	. 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 7184	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
9	6, 7, 8	syl2anc 582	. 2 ⊢ (𝜑 → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
10	5, 9	eqtrd 2765	1 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2098   ∪ cun 3938  {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 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550

This theorem is referenced by:  bj-fvsnun2  36791  bj-fvmptunsn1  36792