Theorem bj-fvmptunsn1 37223

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 2740	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 6272	. . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	4	sseli 4004	. . 3 ⊢ (𝐶 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝐶 ∈ 𝐴)
6	2, 5	nsyl 140	. 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 37220	1 ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2108   ∪ cun 3974  {csn 4648  ⟨cop 4654   ↦ cmpt 5249  dom cdm 5700  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581

This theorem is referenced by:  bj-iomnnom  37225