Theorem bj-fvmptunsn2 36868

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 2849	. . . 4 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐸 = 𝐶)
5	2, 3, 4	syl2anc 582	. . 3 ⊢ (𝜑 → ¬ 𝐸 = 𝐶)
6	1, 5	bj-fununsn1 36863	. 2 ⊢ (𝜑 → (𝐹‘𝐸) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸))
7		eqidd 2726	. . 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 7011	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸) = 𝐺)
11	6, 10	eqtrd 2765	1 ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∪ cun 3942  {csn 4630  ⟨cop 4636   ↦ cmpt 5232  ‘cfv 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557

This theorem is referenced by: (None)