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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‘∅)〉, 〈1o, (𝐺‘1o)〉} = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6683 | . . . 4 ⊢ (𝐺‘∅) ∈ V | |
2 | fvex 6683 | . . . 4 ⊢ (𝐺‘1o) ∈ V | |
3 | fnpr2o 16830 | . . . 4 ⊢ (((𝐺‘∅) ∈ V ∧ (𝐺‘1o) ∈ V) → {〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉} Fn 2o) | |
4 | 1, 2, 3 | mp2an 690 | . . 3 ⊢ {〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉} Fn 2o |
5 | 4 | a1i 11 | . 2 ⊢ (𝐺 Fn 2o → {〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉} Fn 2o) |
6 | id 22 | . 2 ⊢ (𝐺 Fn 2o → 𝐺 Fn 2o) | |
7 | elpri 4589 | . . . . 5 ⊢ (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o)) | |
8 | df2o3 8117 | . . . . 5 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleq2s 2931 | . . . 4 ⊢ (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o)) |
10 | fvpr0o 16832 | . . . . . . 7 ⊢ ((𝐺‘∅) ∈ V → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘∅) = (𝐺‘∅)) | |
11 | 1, 10 | ax-mp 5 | . . . . . 6 ⊢ ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘∅) = (𝐺‘∅) |
12 | fveq2 6670 | . . . . . 6 ⊢ (𝑘 = ∅ → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘∅)) | |
13 | fveq2 6670 | . . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅)) | |
14 | 11, 12, 13 | 3eqtr4a 2882 | . . . . 5 ⊢ (𝑘 = ∅ → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) |
15 | fvpr1o 16833 | . . . . . . 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 6670 | . . . . . 6 ⊢ (𝑘 = 1o → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘1o)) | |
18 | fveq2 6670 | . . . . . 6 ⊢ (𝑘 = 1o → (𝐺‘𝑘) = (𝐺‘1o)) | |
19 | 16, 17, 18 | 3eqtr4a 2882 | . . . . 5 ⊢ (𝑘 = 1o → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) |
20 | 14, 19 | jaoi 853 | . . . 4 ⊢ ((𝑘 = ∅ ∨ 𝑘 = 1o) → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) |
21 | 9, 20 | syl 17 | . . 3 ⊢ (𝑘 ∈ 2o → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) |
22 | 21 | adantl 484 | . 2 ⊢ ((𝐺 Fn 2o ∧ 𝑘 ∈ 2o) → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) |
23 | 5, 6, 22 | eqfnfvd 6805 | 1 ⊢ (𝐺 Fn 2o → {〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉} = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 {cpr 4569 〈cop 4573 Fn wfn 6350 ‘cfv 6355 1oc1o 8095 2oc2o 8096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-om 7581 df-1o 8102 df-2o 8103 |
This theorem is referenced by: xpsff1o 16840 xpstopnlem2 22419 |
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