Theorem bj-fvmptunsn1 34672

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 2798	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 6062	. . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	4	sseli 3911	. . 3 ⊢ (𝐶 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝐶 ∈ 𝐴)
6	2, 5	nsyl 142	. 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 34669	1 ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111   ∪ cun 3879  {csn 4525  ⟨cop 4531   ↦ cmpt 5110  dom cdm 5519  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332

This theorem is referenced by:  bj-iomnnom  34674