Theorem bj-fvmptunsn1 38383

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 2763	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 6256	. . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	4	sseli 3909	. . 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 38380	1 ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  {csn 4588  ⟨cop 4838   ↦ cmpt 5190  dom cdm 5656  ‘cfv 6561

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fv 6569

This theorem is used by:  bj-iomnnom  38385