Theorem fmptap 5852

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 5851	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
4	1, 2, 3	mp2an 426	. . . 4 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)
5		elsni 3691	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
6		fmptap.2	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐵)
7	5, 6	syl 14	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝐶 = 𝐵)
8	7	mpteq2ia 4180	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)
9	4, 8	eqtr4i 2255	. . 3 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)
10	9	uneq2i 3360	. 2 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
11		mptun 5471	. 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12		fmptap.1	. . 3 ⊢ (𝑅 ∪ {𝐴}) = 𝑆
13		mpteq1 4178	. . 3 ⊢ ((𝑅 ∪ {𝐴}) = 𝑆 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶))
14	12, 13	ax-mp 5	. 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)
15	10, 11, 14	3eqtr2i 2258	1 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∪ cun 3199  {csn 3673  ⟨cop 3676   ↦ cmpt 4155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340

This theorem is referenced by: (None)