Theorem xpsfeq 13558

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 6674	. . . 4 ⊢ ∅ ∈ 2o
2		funfvex 5687	. . . . 5 ⊢ ((Fun 𝐺 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ V)
3	2	funfni 5458	. . . 4 ⊢ ((𝐺 Fn 2o ∧ ∅ ∈ 2o) → (𝐺‘∅) ∈ V)
4	1, 3	mpan2 425	. . 3 ⊢ (𝐺 Fn 2o → (𝐺‘∅) ∈ V)
5		1lt2o 6675	. . . 4 ⊢ 1o ∈ 2o
6		funfvex 5687	. . . . 5 ⊢ ((Fun 𝐺 ∧ 1o ∈ dom 𝐺) → (𝐺‘1o) ∈ V)
7	6	funfni 5458	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 1o ∈ 2o) → (𝐺‘1o) ∈ V)
8	5, 7	mpan2 425	. . 3 ⊢ (𝐺 Fn 2o → (𝐺‘1o) ∈ V)
9		fnpr2o 13552	. . 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 3712	. . . 4 ⊢ (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o))
13		df2o3 6662	. . . 4 ⊢ 2o = {∅, 1o}
14	12, 13	eleq2s 2327	. . 3 ⊢ (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o))
15		fvpr0o 13554	. . . . . . 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 5670	. . . . . 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 5670	. . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
21	20	adantl 277	. . . . 5 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → (𝐺‘𝑘) = (𝐺‘∅))
22	17, 19, 21	3eqtr4d 2275	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
23		fvpr1o 13555	. . . . . . 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 5670	. . . . . 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 5670	. . . . . 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 2275	. . . 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 5778	1 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2203  Vcvv 2813  ∅c0 3508  {cpr 3690  ⟨cop 3692   Fn wfn 5347  ‘cfv 5352  1_oc1o 6640  2_oc2o 6641

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-1o 6647  df-2o 6648

This theorem is referenced by:  xpsff1o  13562