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Theorem fmptapd 5275
 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 5273 . . . . 5 ((A V B V) → {⟨A, B⟩} = (x {A} ↦ B))
41, 2, 3syl2anc 393 . . . 4 (φ → {⟨A, B⟩} = (x {A} ↦ B))
5 elsni 3370 . . . . . 6 (x {A} → x = A)
6 fmptapd.2 . . . . . 6 ((φ x = A) → 𝐶 = B)
75, 6sylan2 270 . . . . 5 ((φ x {A}) → 𝐶 = B)
87mpteq2dva 3817 . . . 4 (φ → (x {A} ↦ 𝐶) = (x {A} ↦ B))
94, 8eqtr4d 2053 . . 3 (φ → {⟨A, B⟩} = (x {A} ↦ 𝐶))
109uneq2d 3070 . 2 (φ → ((x 𝑅𝐶) ∪ {⟨A, B⟩}) = ((x 𝑅𝐶) ∪ (x {A} ↦ 𝐶)))
11 mptun 4951 . . 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 3812 . 2 (φ → (x (𝑅 ∪ {A}) ↦ 𝐶) = (x 𝑆𝐶))
1510, 12, 143eqtr2d 2056 1 (φ → ((x 𝑅𝐶) ∪ {⟨A, B⟩}) = (x 𝑆𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  Vcvv 2531   ∪ cun 2888  {csn 3346  ⟨cop 3349   ↦ cmpt 3788 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832 This theorem is referenced by:  fmptpr  5276
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