Theorem xpsfeq 13666

Description: A function on 2_o 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.

Step	Hyp	Ref	Expression
1		0lt2o 6714	. . . 4 ⊢ ∅ ∈ 2o
2		funfvex 5712	. . . . 5 ⊢ ((Fun 𝐺 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ V)
3	2	funfni 5483	. . . 4 ⊢ ((𝐺 Fn 2o ∧ ∅ ∈ 2o) → (𝐺‘∅) ∈ V)
4	1, 3	mpan2 429	. . 3 ⊢ (𝐺 Fn 2o → (𝐺‘∅) ∈ V)
5		1lt2o 6715	. . . 4 ⊢ 1o ∈ 2o
6		funfvex 5712	. . . . 5 ⊢ ((Fun 𝐺 ∧ 1o ∈ dom 𝐺) → (𝐺‘1o) ∈ V)
7	6	funfni 5483	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 1o ∈ 2o) → (𝐺‘1o) ∈ V)
8	5, 7	mpan2 429	. . 3 ⊢ (𝐺 Fn 2o → (𝐺‘1o) ∈ V)
9		fnpr2o 13660	. . 3 ⊢ (((𝐺‘∅) ∈ V ∧ (𝐺‘1o) ∈ V) → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
10	4, 8, 9	syl2anc 415	. 2 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
11		id 19	. 2 ⊢ (𝐺 Fn 2o → 𝐺 Fn 2o)
12		elpri 3732	. . . 4 ⊢ (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o))
13		df2o3 6702	. . . 4 ⊢ 2o = {∅, 1o}
14	12, 13	eleq2s 2333	. . 3 ⊢ (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o))
15		fvpr0o 13662	. . . . . . 7 ⊢ ((𝐺‘∅) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
16	4, 15	syl 14	. . . . . 6 ⊢ (𝐺 Fn 2o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
17	16	adantr 276	. . . . 5 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
18		fveq2 5695	. . . . . 6 ⊢ (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅))
19	18	adantl 277	. . . . 5 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅))
20		fveq2 5695	. . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
21	20	adantl 277	. . . . 5 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → (𝐺‘𝑘) = (𝐺‘∅))
22	17, 19, 21	3eqtr4d 2281	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
23		fvpr1o 13663	. . . . . . 7 ⊢ ((𝐺‘1o) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o))
24	8, 23	syl 14	. . . . . 6 ⊢ (𝐺 Fn 2o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o))
25	24	adantr 276	. . . . 5 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o))
26		fveq2 5695	. . . . . 6 ⊢ (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o))
27	26	adantl 277	. . . . 5 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o))
28		fveq2 5695	. . . . . 6 ⊢ (𝑘 = 1o → (𝐺‘𝑘) = (𝐺‘1o))
29	28	adantl 277	. . . . 5 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = 1o) → (𝐺‘𝑘) = (𝐺‘1o))
30	25, 27, 29	3eqtr4d 2281	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
31	22, 30	jaodan 809	. . 3 ⊢ ((𝐺 Fn 2o ∧ (𝑘 = ∅ ∨ 𝑘 = 1o)) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
32	14, 31	sylan2 286	. 2 ⊢ ((𝐺 Fn 2o ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
33	10, 11, 32	eqfnfvd 5809	1 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  {cpr 3710  ⟨cop 3712   Fn wfn 5372  ‘cfv 5377  1_oc1o 6680  2_oc2o 6681

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-1o 6687  df-2o 6688

This theorem is used by:  xpsff1o  13670