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Theorem fmptap 6477
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 6474 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
41, 2, 3mp2an 708 . . . 4 {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)
5 elsni 4227 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
6 fmptap.2 . . . . . 6 (𝑥 = 𝐴𝐶 = 𝐵)
75, 6syl 17 . . . . 5 (𝑥 ∈ {𝐴} → 𝐶 = 𝐵)
87mpteq2ia 4773 . . . 4 (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)
94, 8eqtr4i 2676 . . 3 {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)
109uneq2i 3797 . 2 ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
11 mptun 6063 . 2 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12 fmptap.1 . . 3 (𝑅 ∪ {𝐴}) = 𝑆
13 mpteq1 4770 . . 3 ((𝑅 ∪ {𝐴}) = 𝑆 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶))
1412, 13ax-mp 5 . 2 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶)
1510, 11, 143eqtr2i 2679 1 ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  {csn 4210  cop 4216  cmpt 4762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933
This theorem is referenced by: (None)
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