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Theorem fmptapd 6320
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 6318 . . 3 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶))
54uneq2d 3729 . 2 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
6 mptun 5924 . . 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 4661 . 2 (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶))
105, 7, 93eqtr2d 2650 1 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cun 3538  {csn 4125  cop 4131  cmpt 4638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4639  df-mpt 4640
This theorem is referenced by:  fmptpr  6321  poimirlem3  32376  poimirlem4  32377  poimirlem16  32389  poimirlem17  32390  poimirlem19  32392  poimirlem20  32393
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