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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 42997 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 42998 | . . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
3 | 2 | eleq2d 2830 | . . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1o}))) |
4 | 3 | adantr 480 | . 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1o}))) |
5 | elmapi 8909 | . . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → 𝑋:𝐴⟶2o) | |
6 | ffn 6749 | . . . 4 ⊢ (𝑋:𝐴⟶2o → 𝑋 Fn 𝐴) | |
7 | fniniseg 7095 | . . . 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 539 | . 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑋‘𝑌) = 1o)) |
10 | 4, 9 | bitrd 279 | 1 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {csn 4648 ↦ cmpt 5249 ◡ccnv 5699 “ cima 5703 Fn wfn 6570 ⟶wf 6571 ‘cfv 6575 (class class class)co 7450 1oc1o 8517 2oc2o 8518 ↑m cmap 8886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1st 8032 df-2nd 8033 df-map 8888 |
This theorem is referenced by: wepwsolem 43001 |
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