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| 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 6919 | . . . 4 ⊢ (𝐺‘∅) ∈ V | |
| 2 | fvex 6919 | . . . 4 ⊢ (𝐺‘1o) ∈ V | |
| 3 | fnpr2o 17602 | . . . 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 4649 | . . . . 5 ⊢ (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o)) | |
| 8 | df2o3 8514 | . . . . 5 ⊢ 2o = {∅, 1o} | |
| 9 | 7, 8 | eleq2s 2859 | . . . 4 ⊢ (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o)) | 
| 10 | fvpr0o 17604 | . . . . . . 7 ⊢ ((𝐺‘∅) ∈ V → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘∅) = (𝐺‘∅)) | |
| 11 | 1, 10 | ax-mp 5 | . . . . . 6 ⊢ ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘∅) = (𝐺‘∅) | 
| 12 | fveq2 6906 | . . . . . 6 ⊢ (𝑘 = ∅ → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘∅)) | |
| 13 | fveq2 6906 | . . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅)) | |
| 14 | 11, 12, 13 | 3eqtr4a 2803 | . . . . 5 ⊢ (𝑘 = ∅ → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) | 
| 15 | fvpr1o 17605 | . . . . . . 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 6906 | . . . . . 6 ⊢ (𝑘 = 1o → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘1o)) | |
| 18 | fveq2 6906 | . . . . . 6 ⊢ (𝑘 = 1o → (𝐺‘𝑘) = (𝐺‘1o)) | |
| 19 | 16, 17, 18 | 3eqtr4a 2803 | . . . . 5 ⊢ (𝑘 = 1o → ({〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉}‘𝑘) = (𝐺‘𝑘)) | 
| 20 | 14, 19 | jaoi 858 | . . . 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 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {cpr 4628 〈cop 4632 Fn wfn 6556 ‘cfv 6561 1oc1o 8499 2oc2o 8500 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-om 7888 df-1o 8506 df-2o 8507 | 
| This theorem is referenced by: xpsff1o 17612 xpstopnlem2 23819 | 
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