Theorem xpsfeq 16224

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‘∅)} +𝑐 {(𝐺‘1𝑜)}) = 𝐺)

Proof of Theorem xpsfeq

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . 4 ⊢ (𝐺‘∅) ∈ V
2		fvex 6201	. . . 4 ⊢ (𝐺‘1𝑜) ∈ V
3		xpscfn 16219	. . . 4 ⊢ (((𝐺‘∅) ∈ V ∧ (𝐺‘1𝑜) ∈ V) → ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) Fn 2𝑜)
4	1, 2, 3	mp2an 708	. . 3 ⊢ ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) Fn 2𝑜
5	4	a1i 11	. 2 ⊢ (𝐺 Fn 2𝑜 → ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) Fn 2𝑜)
6		id 22	. 2 ⊢ (𝐺 Fn 2𝑜 → 𝐺 Fn 2𝑜)
7		elpri 4197	. . . . 5 ⊢ (𝑘 ∈ {∅, 1𝑜} → (𝑘 = ∅ ∨ 𝑘 = 1𝑜))
8		df2o3 7573	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
9	7, 8	eleq2s 2719	. . . 4 ⊢ (𝑘 ∈ 2𝑜 → (𝑘 = ∅ ∨ 𝑘 = 1𝑜))
10		xpsc0 16220	. . . . . . 7 ⊢ ((𝐺‘∅) ∈ V → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘∅) = (𝐺‘∅))
11	1, 10	ax-mp 5	. . . . . 6 ⊢ (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘∅) = (𝐺‘∅)
12		fveq2 6191	. . . . . 6 ⊢ (𝑘 = ∅ → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘∅))
13		fveq2 6191	. . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
14	11, 12, 13	3eqtr4a 2682	. . . . 5 ⊢ (𝑘 = ∅ → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘))
15		xpsc1 16221	. . . . . . 7 ⊢ ((𝐺‘1𝑜) ∈ V → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘1𝑜) = (𝐺‘1𝑜))
16	2, 15	ax-mp 5	. . . . . 6 ⊢ (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘1𝑜) = (𝐺‘1𝑜)
17		fveq2 6191	. . . . . 6 ⊢ (𝑘 = 1𝑜 → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘1𝑜))
18		fveq2 6191	. . . . . 6 ⊢ (𝑘 = 1𝑜 → (𝐺‘𝑘) = (𝐺‘1𝑜))
19	16, 17, 18	3eqtr4a 2682	. . . . 5 ⊢ (𝑘 = 1𝑜 → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘))
20	14, 19	jaoi 394	. . . 4 ⊢ ((𝑘 = ∅ ∨ 𝑘 = 1𝑜) → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘))
21	9, 20	syl 17	. . 3 ⊢ (𝑘 ∈ 2𝑜 → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘))
22	21	adantl 482	. 2 ⊢ ((𝐺 Fn 2𝑜 ∧ 𝑘 ∈ 2𝑜) → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘))
23	5, 6, 22	eqfnfvd 6314	1 ⊢ (𝐺 Fn 2𝑜 → ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) = 𝐺)

Syntax hints:   → wi 4   ∨ wo 383   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177  {cpr 4179  ^◡ccnv 5113   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-cda 8990

This theorem is referenced by:  xpsff1o  16228  xpstopnlem2  21614