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Theorem fmptap 7163
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 = 𝐴𝐶 = 𝐵)
Assertion
Ref Expression
fmptap ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem fmptap
StepHypRef Expression
1 fmptap.0a . . . . 5 𝐴 ∈ V
2 fmptap.0b . . . . 5 𝐵 ∈ V
3 fmptsn 7160 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
41, 2, 3mp2an 689 . . . 4 {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)
5 elsni 4640 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
6 fmptap.2 . . . . . 6 (𝑥 = 𝐴𝐶 = 𝐵)
75, 6syl 17 . . . . 5 (𝑥 ∈ {𝐴} → 𝐶 = 𝐵)
87mpteq2ia 5244 . . . 4 (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)
94, 8eqtr4i 2757 . . 3 {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)
109uneq2i 4155 . 2 ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
11 mptun 6689 . 2 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12 fmptap.1 . . 3 (𝑅 ∪ {𝐴}) = 𝑆
1312mpteq1i 5237 . 2 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶)
1410, 11, 133eqtr2i 2760 1 ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  cun 3941  {csn 4623  cop 4629  cmpt 5224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543
This theorem is referenced by: (None)
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