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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 41405 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 7862 | . . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → ◡𝑋 ∈ V) | |
2 | imaexg 7853 | . . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {1o}) ∈ V) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (◡𝑋 “ {1o}) ∈ V) |
4 | cnveq 5830 | . . . 4 ⊢ (𝑥 = 𝑋 → ◡𝑥 = ◡𝑋) | |
5 | 4 | imaeq1d 6013 | . . 3 ⊢ (𝑥 = 𝑋 → (◡𝑥 “ {1o}) = (◡𝑋 “ {1o})) |
6 | pw2f1o2.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) | |
7 | 5, 6 | fvmptg 6947 | . 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 1542 ∈ wcel 2107 Vcvv 3444 {csn 4587 ↦ cmpt 5189 ◡ccnv 5633 “ cima 5637 ‘cfv 6497 (class class class)co 7358 1oc1o 8406 2oc2o 8407 ↑m cmap 8768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fv 6505 |
This theorem is referenced by: pw2f1o2val2 41407 |
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