Theorem bj-fununsn2 38367

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 4160	. . . 4 ⊢ (𝐺 ∪ {⟨𝐵, 𝐶⟩}) = ({⟨𝐵, 𝐶⟩} ∪ 𝐺)
3	1, 2	eqtrdi 2814	. . 3 ⊢ (𝜑 → 𝐹 = ({⟨𝐵, 𝐶⟩} ∪ 𝐺))
4		bj-fununsn2.neldm	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
5	3, 4	bj-funun 38365	. 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 7208	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
9	6, 7, 8	syl2anc 596	. 2 ⊢ (𝜑 → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
10	5, 9	eqtrd 2798	1 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  {csn 4588  ⟨cop 4838  dom cdm 5655  ‘cfv 6558

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fv 6566

This theorem is used by:  bj-fvsnun2  38369  bj-fvmptunsn1  38370