Theorem fmptap 5686

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 5685	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
4	1, 2, 3	mp2an 424	. . . 4 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)
5		elsni 3601	. . . . . 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 4075	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)
9	4, 8	eqtr4i 2194	. . 3 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)
10	9	uneq2i 3278	. 2 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
11		mptun 5329	. 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12		fmptap.1	. . 3 ⊢ (𝑅 ∪ {𝐴}) = 𝑆
13		mpteq1 4073	. . 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 2197	1 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ∪ cun 3119  {csn 3583  ⟨cop 3586   ↦ cmpt 4050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by: (None)