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Theorem fmptapd 5297
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 ((φ x = A) → 𝐶 = B)
Assertion
Ref Expression
fmptapd (φ → ((x 𝑅𝐶) ∪ {⟨A, B⟩}) = (x 𝑆𝐶))
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 5295 . . . . 5 ((A V B V) → {⟨A, B⟩} = (x {A} ↦ B))
41, 2, 3syl2anc 391 . . . 4 (φ → {⟨A, B⟩} = (x {A} ↦ B))
5 elsni 3391 . . . . . 6 (x {A} → x = A)
6 fmptapd.2 . . . . . 6 ((φ x = A) → 𝐶 = B)
75, 6sylan2 270 . . . . 5 ((φ x {A}) → 𝐶 = B)
87mpteq2dva 3838 . . . 4 (φ → (x {A} ↦ 𝐶) = (x {A} ↦ B))
94, 8eqtr4d 2072 . . 3 (φ → {⟨A, B⟩} = (x {A} ↦ 𝐶))
109uneq2d 3091 . 2 (φ → ((x 𝑅𝐶) ∪ {⟨A, B⟩}) = ((x 𝑅𝐶) ∪ (x {A} ↦ 𝐶)))
11 mptun 4972 . . 3 (x (𝑅 ∪ {A}) ↦ 𝐶) = ((x 𝑅𝐶) ∪ (x {A} ↦ 𝐶))
1211a1i 9 . 2 (φ → (x (𝑅 ∪ {A}) ↦ 𝐶) = ((x 𝑅𝐶) ∪ (x {A} ↦ 𝐶)))
13 fmptapd.1 . . 3 (φ → (𝑅 ∪ {A}) = 𝑆)
1413mpteq1d 3833 . 2 (φ → (x (𝑅 ∪ {A}) ↦ 𝐶) = (x 𝑆𝐶))
1510, 12, 143eqtr2d 2075 1 (φ → ((x 𝑅𝐶) ∪ {⟨A, B⟩}) = (x 𝑆𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cun 2909  {csn 3367  cop 3370  cmpt 3809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  fmptpr  5298
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