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Theorem xpsfeq 16832
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 fvex 6662 . . . 4 (𝐺‘∅) ∈ V
2 fvex 6662 . . . 4 (𝐺‘1o) ∈ V
3 fnpr2o 16826 . . . 4 (((𝐺‘∅) ∈ V ∧ (𝐺‘1o) ∈ V) → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
41, 2, 3mp2an 691 . . 3 {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o
54a1i 11 . 2 (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
6 id 22 . 2 (𝐺 Fn 2o𝐺 Fn 2o)
7 elpri 4550 . . . . 5 (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o))
8 df2o3 8104 . . . . 5 2o = {∅, 1o}
97, 8eleq2s 2911 . . . 4 (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o))
10 fvpr0o 16828 . . . . . . 7 ((𝐺‘∅) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
111, 10ax-mp 5 . . . . . 6 ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅)
12 fveq2 6649 . . . . . 6 (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅))
13 fveq2 6649 . . . . . 6 (𝑘 = ∅ → (𝐺𝑘) = (𝐺‘∅))
1411, 12, 133eqtr4a 2862 . . . . 5 (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
15 fvpr1o 16829 . . . . . . 7 ((𝐺‘1o) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o))
162, 15ax-mp 5 . . . . . 6 ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o)
17 fveq2 6649 . . . . . 6 (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o))
18 fveq2 6649 . . . . . 6 (𝑘 = 1o → (𝐺𝑘) = (𝐺‘1o))
1916, 17, 183eqtr4a 2862 . . . . 5 (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
2014, 19jaoi 854 . . . 4 ((𝑘 = ∅ ∨ 𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
219, 20syl 17 . . 3 (𝑘 ∈ 2o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
2221adantl 485 . 2 ((𝐺 Fn 2o𝑘 ∈ 2o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
235, 6, 22eqfnfvd 6786 1 (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1538  wcel 2112  Vcvv 3444  c0 4246  {cpr 4530  cop 4534   Fn wfn 6323  cfv 6328  1oc1o 8082  2oc2o 8083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-om 7565  df-1o 8089  df-2o 8090
This theorem is referenced by:  xpsff1o  16836  xpstopnlem2  22420
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