Theorem fmptap 7162

Description: Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
fmptap.0a	⊢ 𝐴 ∈ V
fmptap.0b	⊢ 𝐵 ∈ V
fmptap.1	⊢ (𝑅 ∪ {𝐴}) = 𝑆
fmptap.2	⊢ (𝑥 = 𝐴 → 𝐶 = 𝐵)

Assertion

Ref	Expression
fmptap	⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem fmptap

Step	Hyp	Ref	Expression
1		fmptap.0a	. . . . 5 ⊢ 𝐴 ∈ V
2		fmptap.0b	. . . . 5 ⊢ 𝐵 ∈ V
3		fmptsn 7159	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
4	1, 2, 3	mp2an 692	. . . 4 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)
5		elsni 4618	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
6		fmptap.2	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐵)
7	5, 6	syl 17	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝐶 = 𝐵)
8	7	mpteq2ia 5216	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)
9	4, 8	eqtr4i 2761	. . 3 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)
10	9	uneq2i 4140	. 2 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
11		mptun 6684	. 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12		fmptap.1	. . 3 ⊢ (𝑅 ∪ {𝐴}) = 𝑆
13	12	mpteq1i 5211	. 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)
14	10, 11, 13	3eqtr2i 2764	1 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924  {csn 4601  ⟨cop 4607   ↦ cmpt 5201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538

This theorem is referenced by: (None)