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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pw2f1o2val2 | Structured version Visualization version GIF version |
Description: Membership in a mapped set under the pw2f1o2 39979 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
pw2f1o2.f | ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) |
Ref | Expression |
---|---|
pw2f1o2val2 | ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw2f1o2.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) | |
2 | 1 | pw2f1o2val 39980 | . . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
3 | 2 | eleq2d 2875 | . . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1o}))) |
4 | 3 | adantr 484 | . 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1o}))) |
5 | elmapi 8411 | . . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → 𝑋:𝐴⟶2o) | |
6 | ffn 6487 | . . . 4 ⊢ (𝑋:𝐴⟶2o → 𝑋 Fn 𝐴) | |
7 | fniniseg 6807 | . . . 4 ⊢ (𝑋 Fn 𝐴 → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑌 ∈ 𝐴 ∧ (𝑋‘𝑌) = 1o))) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑌 ∈ 𝐴 ∧ (𝑋‘𝑌) = 1o))) |
9 | 8 | baibd 543 | . 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑋‘𝑌) = 1o)) |
10 | 4, 9 | bitrd 282 | 1 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 ↦ cmpt 5110 ◡ccnv 5518 “ cima 5522 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 2oc2o 8079 ↑m cmap 8389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 |
This theorem is referenced by: wepwsolem 39986 |
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