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Theorem xpsfeq 17610
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 6920 . . . 4 (𝐺‘∅) ∈ V
2 fvex 6920 . . . 4 (𝐺‘1o) ∈ V
3 fnpr2o 17604 . . . 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 4654 . . . . 5 (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o))
8 df2o3 8513 . . . . 5 2o = {∅, 1o}
97, 8eleq2s 2857 . . . 4 (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o))
10 fvpr0o 17606 . . . . . . 7 ((𝐺‘∅) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
111, 10ax-mp 5 . . . . . 6 ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅)
12 fveq2 6907 . . . . . 6 (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅))
13 fveq2 6907 . . . . . 6 (𝑘 = ∅ → (𝐺𝑘) = (𝐺‘∅))
1411, 12, 133eqtr4a 2801 . . . . 5 (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
15 fvpr1o 17607 . . . . . . 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 6907 . . . . . 6 (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o))
18 fveq2 6907 . . . . . 6 (𝑘 = 1o → (𝐺𝑘) = (𝐺‘1o))
1916, 17, 183eqtr4a 2801 . . . . 5 (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺𝑘))
2014, 19jaoi 857 . . . 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 847   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {cpr 4633  cop 4637   Fn wfn 6558  cfv 6563  1oc1o 8498  2oc2o 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506
This theorem is referenced by:  xpsff1o  17614  xpstopnlem2  23835
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