Theorem fmptap 7042

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 7039	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
4	1, 2, 3	mp2an 689	. . . 4 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)
5		elsni 4578	. . . . . 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 5177	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)
9	4, 8	eqtr4i 2769	. . 3 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)
10	9	uneq2i 4094	. 2 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
11		mptun 6579	. 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12		fmptap.1	. . 3 ⊢ (𝑅 ∪ {𝐴}) = 𝑆
13	12	mpteq1i 5170	. 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)
14	10, 11, 13	3eqtr2i 2772	1 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885  {csn 4561  ⟨cop 4567   ↦ cmpt 5157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440

This theorem is referenced by: (None)