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Theorem fmptap 7179
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 7176 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
41, 2, 3mp2an 691 . . . 4 {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)
5 elsni 4646 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
6 fmptap.2 . . . . . 6 (𝑥 = 𝐴𝐶 = 𝐵)
75, 6syl 17 . . . . 5 (𝑥 ∈ {𝐴} → 𝐶 = 𝐵)
87mpteq2ia 5251 . . . 4 (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)
94, 8eqtr4i 2759 . . 3 {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)
109uneq2i 4159 . 2 ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
11 mptun 6701 . 2 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12 fmptap.1 . . 3 (𝑅 ∪ {𝐴}) = 𝑆
1312mpteq1i 5244 . 2 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶)
1410, 11, 133eqtr2i 2762 1 ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3471  cun 3945  {csn 4629  cop 4635  cmpt 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555
This theorem is referenced by: (None)
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