Theorem fmptap 5353

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 5352	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
4	1, 2, 3	mp2an 402	. . . 4 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)
5		elsni 3393	. . . . . 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 3843	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)
9	4, 8	eqtr4i 2063	. . 3 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)
10	9	uneq2i 3094	. 2 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
11		mptun 5029	. 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12		fmptap.1	. . 3 ⊢ (𝑅 ∪ {𝐴}) = 𝑆
13		mpteq1 3841	. . 3 ⊢ ((𝑅 ∪ {𝐴}) = 𝑆 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶))
14	12, 13	ax-mp 7	. 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)
15	10, 11, 14	3eqtr2i 2066	1 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915  {csn 3375  ⟨cop 3378   ↦ cmpt 3818

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909

This theorem is referenced by: (None)