Theorem bj-fvmptunsn2 37592

Description: Value of a function expressed as a union of a mapsto expression and a singleton on a couple (with disjoint domain) at a point in the domain of the mapsto construction. (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-fvmptunsn2.el	⊢ (𝜑 → 𝐸 ∈ 𝐴)
bj-fvmptunsn2.ex	⊢ (𝜑 → 𝐺 ∈ 𝑉)
bj-fvmptunsn2.is	⊢ ((𝜑 ∧ 𝑥 = 𝐸) → 𝐵 = 𝐺)

Assertion

Ref	Expression
bj-fvmptunsn2	⊢ (𝜑 → (𝐹‘𝐸) = 𝐺)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐸   𝑥,𝐺

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

Proof of Theorem bj-fvmptunsn2

Step	Hyp	Ref	Expression
1		bj-fvmptunsn.un	. . 3 ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩}))
2		bj-fvmptunsn2.el	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
3		bj-fvmptunsn.nel	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
4		nelneq 2861	. . . 4 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐸 = 𝐶)
5	2, 3, 4	syl2anc 585	. . 3 ⊢ (𝜑 → ¬ 𝐸 = 𝐶)
6	1, 5	bj-fununsn1 37587	. 2 ⊢ (𝜑 → (𝐹‘𝐸) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸))
7		eqidd 2738	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8		bj-fvmptunsn2.is	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐸) → 𝐵 = 𝐺)
9		bj-fvmptunsn2.ex	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
10	7, 8, 2, 9	fvmptd 6951	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸) = 𝐺)
11	6, 10	eqtrd 2772	1 ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∪ cun 3888  {csn 4568  ⟨cop 4574   ↦ cmpt 5167  ‘cfv 6494

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fv 6502

This theorem is referenced by: (None)