Theorem bj-fvmptunsn2 37846

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 2885	. . . 4 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐸 = 𝐶)
5	2, 3, 4	syl2anc 595	. . 3 ⊢ (𝜑 → ¬ 𝐸 = 𝐶)
6	1, 5	bj-fununsn1 37841	. 2 ⊢ (𝜑 → (𝐹‘𝐸) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸))
7		eqidd 2762	. . 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 6997	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸) = 𝐺)
11	6, 10	eqtrd 2796	1 ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ∪ cun 3902  {csn 4588  ⟨cop 4594   ↦ cmpt 5191  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544

This theorem is referenced by: (None)