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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 6920 | . . . 4 ⊢ (𝐺‘∅) ∈ V | |
2 | fvex 6920 | . . . 4 ⊢ (𝐺‘1o) ∈ V | |
3 | fnpr2o 17604 | . . . 4 ⊢ (((𝐺‘∅) ∈ V ∧ (𝐺‘1o) ∈ V) → {〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉} Fn 2o) | |
4 | 1, 2, 3 | mp2an 692 | . . 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 4654 | . . . . 5 ⊢ (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o)) | |
8 | df2o3 8513 | . . . . 5 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleq2s 2857 | . . . 4 ⊢ (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o)) |
10 | fvpr0o 17606 | . . . . . . 7 ⊢ ((𝐺‘∅) ∈ V → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘∅) = (𝐺‘∅)) | |
11 | 1, 10 | ax-mp 5 | . . . . . 6 ⊢ ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘∅) = (𝐺‘∅) |
12 | fveq2 6907 | . . . . . 6 ⊢ (𝑘 = ∅ → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘∅)) | |
13 | fveq2 6907 | . . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅)) | |
14 | 11, 12, 13 | 3eqtr4a 2801 | . . . . 5 ⊢ (𝑘 = ∅ → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) |
15 | fvpr1o 17607 | . . . . . . 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 6907 | . . . . . 6 ⊢ (𝑘 = 1o → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘1o)) | |
18 | fveq2 6907 | . . . . . 6 ⊢ (𝑘 = 1o → (𝐺‘𝑘) = (𝐺‘1o)) | |
19 | 16, 17, 18 | 3eqtr4a 2801 | . . . . 5 ⊢ (𝑘 = 1o → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) |
20 | 14, 19 | jaoi 857 | . . . 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 481 | . 2 ⊢ ((𝐺 Fn 2o ∧ 𝑘 ∈ 2o) → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) |
23 | 5, 6, 22 | eqfnfvd 7054 | 1 ⊢ (𝐺 Fn 2o → {〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉} = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {cpr 4633 〈cop 4637 Fn wfn 6558 ‘cfv 6563 1oc1o 8498 2oc2o 8499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-om 7888 df-1o 8505 df-2o 8506 |
This theorem is referenced by: xpsff1o 17614 xpstopnlem2 23835 |
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