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Mirrors > Home > MPE Home > Th. List > Mathboxes > pw2f1o2val2 | Structured version Visualization version GIF version |
Description: Membership in a mapped set under the pw2f1o2 36623 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 |
---|---|
pw2f1o2val2 | ⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1𝑜)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw2f1o2.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) | |
2 | 1 | pw2f1o2val 36624 | . . . 4 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1𝑜})) |
3 | 2 | eleq2d 2673 | . . 3 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1𝑜}))) |
4 | 3 | adantr 480 | . 2 ⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1𝑜}))) |
5 | elmapi 7765 | . . . 4 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → 𝑋:𝐴⟶2𝑜) | |
6 | ffn 5958 | . . . 4 ⊢ (𝑋:𝐴⟶2𝑜 → 𝑋 Fn 𝐴) | |
7 | fniniseg 6246 | . . . 4 ⊢ (𝑋 Fn 𝐴 → (𝑌 ∈ (◡𝑋 “ {1𝑜}) ↔ (𝑌 ∈ 𝐴 ∧ (𝑋‘𝑌) = 1𝑜))) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → (𝑌 ∈ (◡𝑋 “ {1𝑜}) ↔ (𝑌 ∈ 𝐴 ∧ (𝑋‘𝑌) = 1𝑜))) |
9 | 8 | baibd 946 | . 2 ⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑋 “ {1𝑜}) ↔ (𝑋‘𝑌) = 1𝑜)) |
10 | 4, 9 | bitrd 267 | 1 ⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1𝑜)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 2𝑜c2o 7441 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 |
This theorem is referenced by: wepwsolem 36630 |
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