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Mirrors > Home > MPE Home > Th. List > Mathboxes > pw2f1o2val | Structured version Visualization version GIF version |
Description: Function value of the pw2f1o2 37605 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 7112 | . . 3 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → ◡𝑋 ∈ V) | |
2 | imaexg 7103 | . . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {1𝑜}) ∈ V) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → (◡𝑋 “ {1𝑜}) ∈ V) |
4 | cnveq 5296 | . . . 4 ⊢ (𝑥 = 𝑋 → ◡𝑥 = ◡𝑋) | |
5 | 4 | imaeq1d 5465 | . . 3 ⊢ (𝑥 = 𝑋 → (◡𝑥 “ {1𝑜}) = (◡𝑋 “ {1𝑜})) |
6 | pw2f1o2.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) | |
7 | 5, 6 | fvmptg 6280 | . 2 ⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ (◡𝑋 “ {1𝑜}) ∈ V) → (𝐹‘𝑋) = (◡𝑋 “ {1𝑜})) |
8 | 3, 7 | mpdan 702 | 1 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1𝑜})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 ↦ cmpt 4729 ◡ccnv 5113 “ cima 5117 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 2𝑜c2o 7554 ↑𝑚 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 |
This theorem is referenced by: pw2f1o2val2 37607 |
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