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Theorem xpsfeq 17608
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 6919 . . . 4 (𝐺‘∅) ∈ V
2 fvex 6919 . . . 4 (𝐺‘1o) ∈ V
3 fnpr2o 17602 . . . 4 (((𝐺‘∅) ∈ V ∧ (𝐺‘1o) ∈ V) → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
41, 2, 3mp2an 692 . . 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 4649 . . . . 5 (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o))
8 df2o3 8514 . . . . 5 2o = {∅, 1o}
97, 8eleq2s 2859 . . . 4 (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o))
10 fvpr0o 17604 . . . . . . 7 ((𝐺‘∅) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
111, 10ax-mp 5 . . . . . 6 ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅)
12 fveq2 6906 . . . . . 6 (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅))
13 fveq2 6906 . . . . . 6 (𝑘 = ∅ → (𝐺𝑘) = (𝐺‘∅))
1411, 12, 133eqtr4a 2803 . . . . 5 (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
15 fvpr1o 17605 . . . . . . 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 6906 . . . . . 6 (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o))
18 fveq2 6906 . . . . . 6 (𝑘 = 1o → (𝐺𝑘) = (𝐺‘1o))
1916, 17, 183eqtr4a 2803 . . . . 5 (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
2014, 19jaoi 858 . . . 4 ((𝑘 = ∅ ∨ 𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
219, 20syl 17 . . 3 (𝑘 ∈ 2o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
2221adantl 481 . 2 ((𝐺 Fn 2o𝑘 ∈ 2o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
235, 6, 22eqfnfvd 7054 1 (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 848   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  {cpr 4628  cop 4632   Fn wfn 6556  cfv 6561  1oc1o 8499  2oc2o 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507
This theorem is referenced by:  xpsff1o  17612  xpstopnlem2  23819
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