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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 42360 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 7914 | . . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → ◡𝑋 ∈ V) | |
2 | imaexg 7903 | . . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {1o}) ∈ V) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (◡𝑋 “ {1o}) ∈ V) |
4 | cnveq 5867 | . . . 4 ⊢ (𝑥 = 𝑋 → ◡𝑥 = ◡𝑋) | |
5 | 4 | imaeq1d 6052 | . . 3 ⊢ (𝑥 = 𝑋 → (◡𝑥 “ {1o}) = (◡𝑋 “ {1o})) |
6 | pw2f1o2.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) | |
7 | 5, 6 | fvmptg 6990 | . 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ (◡𝑋 “ {1o}) ∈ V) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
8 | 3, 7 | mpdan 684 | 1 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 ↦ cmpt 5224 ◡ccnv 5668 “ cima 5672 ‘cfv 6537 (class class class)co 7405 1oc1o 8460 2oc2o 8461 ↑m cmap 8822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fv 6545 |
This theorem is referenced by: pw2f1o2val2 42362 |
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