Theorem bj-fvmptunsn2 35426

Description: Value of a function expressed as a union of a mapsto expression and a singleton on a couple (with disjoint domain) at a point in the domain of the mapsto construction. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-fvmptunsn.un	⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩}))
bj-fvmptunsn.nel	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
bj-fvmptunsn2.el	⊢ (𝜑 → 𝐸 ∈ 𝐴)
bj-fvmptunsn2.ex	⊢ (𝜑 → 𝐺 ∈ 𝑉)
bj-fvmptunsn2.is	⊢ ((𝜑 ∧ 𝑥 = 𝐸) → 𝐵 = 𝐺)

Assertion

Ref	Expression
bj-fvmptunsn2	⊢ (𝜑 → (𝐹‘𝐸) = 𝐺)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐸   𝑥,𝐺

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem bj-fvmptunsn2

Step	Hyp	Ref	Expression
1		bj-fvmptunsn.un	. . 3 ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩}))
2		bj-fvmptunsn2.el	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
3		bj-fvmptunsn.nel	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
4		nelneq 2863	. . . 4 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐸 = 𝐶)
5	2, 3, 4	syl2anc 584	. . 3 ⊢ (𝜑 → ¬ 𝐸 = 𝐶)
6	1, 5	bj-fununsn1 35421	. 2 ⊢ (𝜑 → (𝐹‘𝐸) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸))
7		eqidd 2739	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8		bj-fvmptunsn2.is	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐸) → 𝐵 = 𝐺)
9		bj-fvmptunsn2.ex	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
10	7, 8, 2, 9	fvmptd 6884	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸) = 𝐺)
11	6, 10	eqtrd 2778	1 ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∪ cun 3886  {csn 4563  ⟨cop 4569   ↦ cmpt 5159  ‘cfv 6435

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6393  df-fun 6437  df-fv 6443

This theorem is referenced by: (None)