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Mirrors > Home > MPE Home > Th. List > Mathboxes > pw2f1o2val | Structured version Visualization version GIF version |
Description: Function value of the pw2f1o2 38448 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 7374 | . . 3 ⊢ (𝑋 ∈ (2o ↑𝑚 𝐴) → ◡𝑋 ∈ V) | |
2 | imaexg 7365 | . . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {1o}) ∈ V) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑋 ∈ (2o ↑𝑚 𝐴) → (◡𝑋 “ {1o}) ∈ V) |
4 | cnveq 5528 | . . . 4 ⊢ (𝑥 = 𝑋 → ◡𝑥 = ◡𝑋) | |
5 | 4 | imaeq1d 5706 | . . 3 ⊢ (𝑥 = 𝑋 → (◡𝑥 “ {1o}) = (◡𝑋 “ {1o})) |
6 | pw2f1o2.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) | |
7 | 5, 6 | fvmptg 6527 | . 2 ⊢ ((𝑋 ∈ (2o ↑𝑚 𝐴) ∧ (◡𝑋 “ {1o}) ∈ V) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
8 | 3, 7 | mpdan 680 | 1 ⊢ (𝑋 ∈ (2o ↑𝑚 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3414 {csn 4397 ↦ cmpt 4952 ◡ccnv 5341 “ cima 5345 ‘cfv 6123 (class class class)co 6905 1oc1o 7819 2oc2o 7820 ↑𝑚 cmap 8122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fv 6131 |
This theorem is referenced by: pw2f1o2val2 38450 |
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