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Mirrors > Home > MPE Home > Th. List > xpsfeq | Structured version Visualization version GIF version |
Description: A function on 2𝑜 is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
xpsfeq | ⊢ (𝐺 Fn 2𝑜 → ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6354 | . . . 4 ⊢ (𝐺‘∅) ∈ V | |
2 | fvex 6354 | . . . 4 ⊢ (𝐺‘1𝑜) ∈ V | |
3 | xpscfn 16413 | . . . 4 ⊢ (((𝐺‘∅) ∈ V ∧ (𝐺‘1𝑜) ∈ V) → ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) Fn 2𝑜) | |
4 | 1, 2, 3 | mp2an 710 | . . 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 4334 | . . . . 5 ⊢ (𝑘 ∈ {∅, 1𝑜} → (𝑘 = ∅ ∨ 𝑘 = 1𝑜)) | |
8 | df2o3 7734 | . . . . 5 ⊢ 2𝑜 = {∅, 1𝑜} | |
9 | 7, 8 | eleq2s 2849 | . . . 4 ⊢ (𝑘 ∈ 2𝑜 → (𝑘 = ∅ ∨ 𝑘 = 1𝑜)) |
10 | xpsc0 16414 | . . . . . . 7 ⊢ ((𝐺‘∅) ∈ V → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘∅) = (𝐺‘∅)) | |
11 | 1, 10 | ax-mp 5 | . . . . . 6 ⊢ (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘∅) = (𝐺‘∅) |
12 | fveq2 6344 | . . . . . 6 ⊢ (𝑘 = ∅ → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘∅)) | |
13 | fveq2 6344 | . . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅)) | |
14 | 11, 12, 13 | 3eqtr4a 2812 | . . . . 5 ⊢ (𝑘 = ∅ → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘)) |
15 | xpsc1 16415 | . . . . . . 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 6344 | . . . . . 6 ⊢ (𝑘 = 1𝑜 → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘1𝑜)) | |
18 | fveq2 6344 | . . . . . 6 ⊢ (𝑘 = 1𝑜 → (𝐺‘𝑘) = (𝐺‘1𝑜)) | |
19 | 16, 17, 18 | 3eqtr4a 2812 | . . . . 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 473 | . 2 ⊢ ((𝐺 Fn 2𝑜 ∧ 𝑘 ∈ 2𝑜) → (◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)})‘𝑘) = (𝐺‘𝑘)) |
23 | 5, 6, 22 | eqfnfvd 6469 | 1 ⊢ (𝐺 Fn 2𝑜 → ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 = wceq 1624 ∈ wcel 2131 Vcvv 3332 ∅c0 4050 {csn 4313 {cpr 4315 ◡ccnv 5257 Fn wfn 6036 ‘cfv 6041 (class class class)co 6805 1𝑜c1o 7714 2𝑜c2o 7715 +𝑐 ccda 9173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-ord 5879 df-on 5880 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-1o 7721 df-2o 7722 df-cda 9174 |
This theorem is referenced by: xpsff1o 16422 xpstopnlem2 21808 |
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