Theorem bj-fvmptunsn1 37761

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 2765	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 6232	. . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	4	sseli 3935	. . 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 37758	1 ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1563   ∈ wcel 2145   ∪ cun 3905  {csn 4585  ⟨cop 4591   ↦ cmpt 5186  dom cdm 5652  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fv 6533

This theorem is referenced by:  bj-iomnnom  37763