Theorem xpsfeq 12770

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 6445	. . . 4 ⊢ ∅ ∈ 2o
2		funfvex 5534	. . . . 5 ⊢ ((Fun 𝐺 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ V)
3	2	funfni 5318	. . . 4 ⊢ ((𝐺 Fn 2o ∧ ∅ ∈ 2o) → (𝐺‘∅) ∈ V)
4	1, 3	mpan2 425	. . 3 ⊢ (𝐺 Fn 2o → (𝐺‘∅) ∈ V)
5		1lt2o 6446	. . . 4 ⊢ 1o ∈ 2o
6		funfvex 5534	. . . . 5 ⊢ ((Fun 𝐺 ∧ 1o ∈ dom 𝐺) → (𝐺‘1o) ∈ V)
7	6	funfni 5318	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 1o ∈ 2o) → (𝐺‘1o) ∈ V)
8	5, 7	mpan2 425	. . 3 ⊢ (𝐺 Fn 2o → (𝐺‘1o) ∈ V)
9		fnpr2o 12764	. . 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 3617	. . . 4 ⊢ (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o))
13		df2o3 6434	. . . 4 ⊢ 2o = {∅, 1o}
14	12, 13	eleq2s 2272	. . 3 ⊢ (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o))
15		fvpr0o 12766	. . . . . . 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 5517	. . . . . 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 5517	. . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
21	20	adantl 277	. . . . 5 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → (𝐺‘𝑘) = (𝐺‘∅))
22	17, 19, 21	3eqtr4d 2220	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = ∅) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
23		fvpr1o 12767	. . . . . . 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 5517	. . . . . 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 5517	. . . . . 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 2220	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
31	22, 30	jaodan 797	. . 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 5619	1 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708   = wceq 1353   ∈ wcel 2148  Vcvv 2739  ∅c0 3424  {cpr 3595  ⟨cop 3597   Fn wfn 5213  ‘cfv 5218  1_oc1o 6413  2_oc2o 6414

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-1o 6420  df-2o 6421

This theorem is referenced by:  xpsff1o  12774