Theorem xpsfeq 13609

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 6687	. . . 4 ⊢ ∅ ∈ 2o
2		funfvex 5692	. . . . 5 ⊢ ((Fun 𝐺 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ V)
3	2	funfni 5463	. . . 4 ⊢ ((𝐺 Fn 2o ∧ ∅ ∈ 2o) → (𝐺‘∅) ∈ V)
4	1, 3	mpan2 425	. . 3 ⊢ (𝐺 Fn 2o → (𝐺‘∅) ∈ V)
5		1lt2o 6688	. . . 4 ⊢ 1o ∈ 2o
6		funfvex 5692	. . . . 5 ⊢ ((Fun 𝐺 ∧ 1o ∈ dom 𝐺) → (𝐺‘1o) ∈ V)
7	6	funfni 5463	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 1o ∈ 2o) → (𝐺‘1o) ∈ V)
8	5, 7	mpan2 425	. . 3 ⊢ (𝐺 Fn 2o → (𝐺‘1o) ∈ V)
9		fnpr2o 13603	. . 3 ⊢ (((𝐺‘∅) ∈ V ∧ (𝐺‘1o) ∈ V) → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
10	4, 8, 9	syl2anc 411	. 2 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
11		id 19	. 2 ⊢ (𝐺 Fn 2o → 𝐺 Fn 2o)
12		elpri 3717	. . . 4 ⊢ (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o))
13		df2o3 6675	. . . 4 ⊢ 2o = {∅, 1o}
14	12, 13	eleq2s 2329	. . 3 ⊢ (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o))
15		fvpr0o 13605	. . . . . . 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 5675	. . . . . 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 5675	. . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
21	20	adantl 277	. . . . 5 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → (𝐺‘𝑘) = (𝐺‘∅))
22	17, 19, 21	3eqtr4d 2277	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
23		fvpr1o 13606	. . . . . . 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 5675	. . . . . 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 5675	. . . . . 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 2277	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
31	22, 30	jaodan 805	. . 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 5783	1 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ∅c0 3512  {cpr 3695  ⟨cop 3697   Fn wfn 5352  ‘cfv 5357  1_oc1o 6653  2_oc2o 6654

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-1o 6660  df-2o 6661

This theorem is referenced by:  xpsff1o  13613