Theorem xpsfeq 16047

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 6113	. . . 4 ⊢ (𝐺‘∅) ∈ V
2		fvex 6113	. . . 4 ⊢ (𝐺‘1𝑜) ∈ V
3		xpscfn 16042	. . . 4 ⊢ (((𝐺‘∅) ∈ V ∧ (𝐺‘1𝑜) ∈ V) → ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) Fn 2𝑜)
4	1, 2, 3	mp2an 704	. . 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 4145	. . . . 5 ⊢ (𝑘 ∈ {∅, 1𝑜} → (𝑘 = ∅ ∨ 𝑘 = 1𝑜))
8		df2o3 7460	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
9	7, 8	eleq2s 2706	. . . 4 ⊢ (𝑘 ∈ 2𝑜 → (𝑘 = ∅ ∨ 𝑘 = 1𝑜))
10		xpsc0 16043	. . . . . . 7 ⊢ ((𝐺‘∅) ∈ V → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘∅) = (𝐺‘∅))
11	1, 10	ax-mp 5	. . . . . 6 ⊢ (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘∅) = (𝐺‘∅)
12		fveq2 6103	. . . . . 6 ⊢ (𝑘 = ∅ → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘∅))
13		fveq2 6103	. . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
14	11, 12, 13	3eqtr4a 2670	. . . . 5 ⊢ (𝑘 = ∅ → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘))
15		xpsc1 16044	. . . . . . 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 6103	. . . . . 6 ⊢ (𝑘 = 1𝑜 → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘1𝑜))
18		fveq2 6103	. . . . . 6 ⊢ (𝑘 = 1𝑜 → (𝐺‘𝑘) = (𝐺‘1𝑜))
19	16, 17, 18	3eqtr4a 2670	. . . . 5 ⊢ (𝑘 = 1𝑜 → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘))
20	14, 19	jaoi 393	. . . 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 481	. 2 ⊢ ((𝐺 Fn 2𝑜 ∧ 𝑘 ∈ 2𝑜) → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘))
23	5, 6, 22	eqfnfvd 6222	1 ⊢ (𝐺 Fn 2𝑜 → ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) = 𝐺)

Syntax hints:   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  {cpr 4127  ^◡ccnv 5037   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441   +_𝑐 ccda 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-2o 7448  df-cda 8873

This theorem is referenced by:  xpsff1o  16051  xpstopnlem2  21424