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Theorem fmptapd 6925
Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
Hypotheses
Ref Expression
fmptapd.0a (𝜑𝐴 ∈ V)
fmptapd.0b (𝜑𝐵 ∈ V)
fmptapd.1 (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
fmptapd.2 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)
Assertion
Ref Expression
fmptapd (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆   𝜑,𝑥
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem fmptapd
StepHypRef Expression
1 fmptapd.2 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)
2 fmptapd.0a . . . 4 (𝜑𝐴 ∈ V)
3 fmptapd.0b . . . 4 (𝜑𝐵 ∈ V)
41, 2, 3fmptsnd 6923 . . 3 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶))
54uneq2d 4136 . 2 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
6 mptun 6487 . . 3 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
76a1i 11 . 2 (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
8 fmptapd.1 . . 3 (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
98mpteq1d 5146 . 2 (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶))
105, 7, 93eqtr2d 2859 1 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  {csn 4557  cop 4563  cmpt 5137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-mpt 5138
This theorem is referenced by:  fmptpr  6926  poimirlem3  34776  poimirlem4  34777  poimirlem16  34789  poimirlem17  34790  poimirlem19  34792  poimirlem20  34793
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