Theorem bj-fununsn2 37242

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 4121	. . . 4 ⊢ (𝐺 ∪ {⟨𝐵, 𝐶⟩}) = ({⟨𝐵, 𝐶⟩} ∪ 𝐺)
3	1, 2	eqtrdi 2780	. . 3 ⊢ (𝜑 → 𝐹 = ({⟨𝐵, 𝐶⟩} ∪ 𝐺))
4		bj-fununsn2.neldm	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
5	3, 4	bj-funun 37240	. 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 7154	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
9	6, 7, 8	syl2anc 584	. 2 ⊢ (𝜑 → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
10	5, 9	eqtrd 2764	1 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∪ cun 3912  {csn 4589  ⟨cop 4595  dom cdm 5638  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519

This theorem is referenced by:  bj-fvsnun2  37244  bj-fvmptunsn1  37245