Theorem bj-fvmptunsn2 34542

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 2939	. . . 4 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐸 = 𝐶)
5	2, 3, 4	syl2anc 586	. . 3 ⊢ (𝜑 → ¬ 𝐸 = 𝐶)
6	1, 5	bj-fununsn1 34537	. 2 ⊢ (𝜑 → (𝐹‘𝐸) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸))
7		eqidd 2824	. . 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 6777	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸) = 𝐺)
11	6, 10	eqtrd 2858	1 ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∪ cun 3936  {csn 4569  ⟨cop 4575   ↦ cmpt 5148  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fv 6365

This theorem is referenced by: (None)