Theorem xpsfeq 17612

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		fvex 6894	. . . 4 ⊢ (𝐺‘∅) ∈ V
2		fvex 6894	. . . 4 ⊢ (𝐺‘1o) ∈ V
3		fnpr2o 17606	. . . 4 ⊢ (((𝐺‘∅) ∈ V ∧ (𝐺‘1o) ∈ V) → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
4	1, 2, 3	mp2an 704	. . 3 ⊢ {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o
5	4	a1i 11	. 2 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o)
6		id 23	. 2 ⊢ (𝐺 Fn 2o → 𝐺 Fn 2o)
7		elpri 4613	. . . . 5 ⊢ (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o))
8		df2o3 8457	. . . . 5 ⊢ 2o = {∅, 1o}
9	7, 8	eleq2s 2881	. . . 4 ⊢ (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o))
10		fvpr0o 17608	. . . . . . 7 ⊢ ((𝐺‘∅) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅))
11	1, 10	ax-mp 5	. . . . . 6 ⊢ ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅)
12		fveq2 6881	. . . . . 6 ⊢ (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅))
13		fveq2 6881	. . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
14	11, 12, 13	3eqtr4a 2824	. . . . 5 ⊢ (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
15		fvpr1o 17609	. . . . . . 7 ⊢ ((𝐺‘1o) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o))
16	2, 15	ax-mp 5	. . . . . 6 ⊢ ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o)
17		fveq2 6881	. . . . . 6 ⊢ (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o))
18		fveq2 6881	. . . . . 6 ⊢ (𝑘 = 1o → (𝐺‘𝑘) = (𝐺‘1o))
19	16, 17, 18	3eqtr4a 2824	. . . . 5 ⊢ (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
20	14, 19	jaoi 870	. . . 4 ⊢ ((𝑘 = ∅ ∨ 𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
21	9, 20	syl 18	. . 3 ⊢ (𝑘 ∈ 2o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
22	21	adantl 486	. 2 ⊢ ((𝐺 Fn 2o ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘))
23	5, 6, 22	eqfnfvd 7028	1 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  {cpr 4591  ⟨cop 4595   Fn wfn 6531  ‘cfv 6536  1_oc1o 8442  2_oc2o 8443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-om 7859  df-1o 8449  df-2o 8450

This theorem is referenced by:  xpsff1o  17616  xpstopnlem2  23968