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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pw2f1o2val | Structured version Visualization version GIF version |
Description: Function value of 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 |
---|---|
pw2f1o2val | ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvexg 7611 | . . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → ◡𝑋 ∈ V) | |
2 | imaexg 7602 | . . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {1o}) ∈ V) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (◡𝑋 “ {1o}) ∈ V) |
4 | cnveq 5708 | . . . 4 ⊢ (𝑥 = 𝑋 → ◡𝑥 = ◡𝑋) | |
5 | 4 | imaeq1d 5895 | . . 3 ⊢ (𝑥 = 𝑋 → (◡𝑥 “ {1o}) = (◡𝑋 “ {1o})) |
6 | pw2f1o2.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) | |
7 | 5, 6 | fvmptg 6743 | . 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ (◡𝑋 “ {1o}) ∈ V) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
8 | 3, 7 | mpdan 686 | 1 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 ↦ cmpt 5110 ◡ccnv 5518 “ cima 5522 ‘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-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 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-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-fv 6332 |
This theorem is referenced by: pw2f1o2val2 39981 |
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