Theorem bj-fvmptunsn1 37934

Description: Value of a function expressed as a union of a mapsto expression 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-fvmptunsn.un	⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩}))
bj-fvmptunsn.nel	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
bj-fvmptunsn1.ex1	⊢ (𝜑 → 𝐶 ∈ 𝑉)
bj-fvmptunsn1.ex2	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
bj-fvmptunsn1	⊢ (𝜑 → (𝐹‘𝐶) = 𝐷)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem bj-fvmptunsn1

Step	Hyp	Ref	Expression
1		bj-fvmptunsn.un	. 2 ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩}))
2		bj-fvmptunsn.nel	. . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
3		eqid 2766	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 6247	. . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	4	sseli 3936	. . 3 ⊢ (𝐶 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝐶 ∈ 𝐴)
6	2, 5	nsyl 141	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵))
7		bj-fvmptunsn1.ex1	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
8		bj-fvmptunsn1.ex2	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
9	1, 6, 7, 8	bj-fununsn2 37931	1 ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146   ∪ cun 3906  {csn 4594  ⟨cop 4600   ↦ cmpt 5197  dom cdm 5666  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551

This theorem is used by:  bj-iomnnom  37936