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Theorem xpsfeq 12833
Description: A function on 2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)
Assertion
Ref Expression
xpsfeq (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)

Proof of Theorem xpsfeq
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 0lt2o 6470 . . . 4 ∅ ∈ 2o
2 funfvex 5554 . . . . 5 ((Fun 𝐺 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ V)
32funfni 5338 . . . 4 ((𝐺 Fn 2o ∧ ∅ ∈ 2o) → (𝐺‘∅) ∈ V)
41, 3mpan2 425 . . 3 (𝐺 Fn 2o → (𝐺‘∅) ∈ V)
5 1lt2o 6471 . . . 4 1o ∈ 2o
6 funfvex 5554 . . . . 5 ((Fun 𝐺 ∧ 1o ∈ dom 𝐺) → (𝐺‘1o) ∈ V)
76funfni 5338 . . . 4 ((𝐺 Fn 2o ∧ 1o ∈ 2o) → (𝐺‘1o) ∈ V)
85, 7mpan2 425 . . 3 (𝐺 Fn 2o → (𝐺‘1o) ∈ V)
9 fnpr2o 12827 . . 3 (((𝐺‘∅) ∈ V ∧ (𝐺‘1o) ∈ V) → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
104, 8, 9syl2anc 411 . 2 (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
11 id 19 . 2 (𝐺 Fn 2o𝐺 Fn 2o)
12 elpri 3633 . . . 4 (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o))
13 df2o3 6459 . . . 4 2o = {∅, 1o}
1412, 13eleq2s 2284 . . 3 (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o))
15 fvpr0o 12829 . . . . . . 7 ((𝐺‘∅) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
164, 15syl 14 . . . . . 6 (𝐺 Fn 2o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
1716adantr 276 . . . . 5 ((𝐺 Fn 2o𝑘 = ∅) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
18 fveq2 5537 . . . . . 6 (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅))
1918adantl 277 . . . . 5 ((𝐺 Fn 2o𝑘 = ∅) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅))
20 fveq2 5537 . . . . . 6 (𝑘 = ∅ → (𝐺𝑘) = (𝐺‘∅))
2120adantl 277 . . . . 5 ((𝐺 Fn 2o𝑘 = ∅) → (𝐺𝑘) = (𝐺‘∅))
2217, 19, 213eqtr4d 2232 . . . 4 ((𝐺 Fn 2o𝑘 = ∅) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
23 fvpr1o 12830 . . . . . . 7 ((𝐺‘1o) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o))
248, 23syl 14 . . . . . 6 (𝐺 Fn 2o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o))
2524adantr 276 . . . . 5 ((𝐺 Fn 2o𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o))
26 fveq2 5537 . . . . . 6 (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o))
2726adantl 277 . . . . 5 ((𝐺 Fn 2o𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o))
28 fveq2 5537 . . . . . 6 (𝑘 = 1o → (𝐺𝑘) = (𝐺‘1o))
2928adantl 277 . . . . 5 ((𝐺 Fn 2o𝑘 = 1o) → (𝐺𝑘) = (𝐺‘1o))
3025, 27, 293eqtr4d 2232 . . . 4 ((𝐺 Fn 2o𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
3122, 30jaodan 798 . . 3 ((𝐺 Fn 2o ∧ (𝑘 = ∅ ∨ 𝑘 = 1o)) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
3214, 31sylan2 286 . 2 ((𝐺 Fn 2o𝑘 ∈ 2o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
3310, 11, 32eqfnfvd 5640 1 (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709   = wceq 1364  wcel 2160  Vcvv 2752  c0 3437  {cpr 3611  cop 3613   Fn wfn 5233  cfv 5238  1oc1o 6438  2oc2o 6439
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-in1 615  ax-in2 616  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-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-res 4659  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246  df-1o 6445  df-2o 6446
This theorem is referenced by:  xpsff1o  12837
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