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Theorem fmptapd 5749
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.0a . . . . 5 (𝜑𝐴 ∈ V)
2 fmptapd.0b . . . . 5 (𝜑𝐵 ∈ V)
3 fmptsn 5747 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
41, 2, 3syl2anc 411 . . . 4 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
5 elsni 3636 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
6 fmptapd.2 . . . . . 6 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)
75, 6sylan2 286 . . . . 5 ((𝜑𝑥 ∈ {𝐴}) → 𝐶 = 𝐵)
87mpteq2dva 4119 . . . 4 (𝜑 → (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵))
94, 8eqtr4d 2229 . . 3 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶))
109uneq2d 3313 . 2 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
11 mptun 5385 . . 3 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
1211a1i 9 . 2 (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
13 fmptapd.1 . . 3 (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
1413mpteq1d 4114 . 2 (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶))
1510, 12, 143eqtr2d 2232 1 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2164  Vcvv 2760  cun 3151  {csn 3618  cop 3621  cmpt 4090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261
This theorem is referenced by:  fmptpr  5750
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