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Mirrors > Home > MPE Home > Th. List > Mathboxes > pwfi2en | Structured version Visualization version GIF version |
Description: Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.) |
Ref | Expression |
---|---|
pwfi2en.s | ⊢ 𝑆 = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} |
Ref | Expression |
---|---|
pwfi2en | ⊢ (𝐴 ∈ 𝑉 → 𝑆 ≈ (𝒫 𝐴 ∩ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwfi2en.s | . . 3 ⊢ 𝑆 = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} | |
2 | eqid 2610 | . . 3 ⊢ (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜})) = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜})) | |
3 | 1, 2 | pwfi2f1o 36684 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜})):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)) |
4 | ovex 6577 | . . . 4 ⊢ (2𝑜 ↑𝑚 𝐴) ∈ V | |
5 | 1, 4 | rabex2 4742 | . . 3 ⊢ 𝑆 ∈ V |
6 | 5 | f1oen 7862 | . 2 ⊢ ((𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜})):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin) → 𝑆 ≈ (𝒫 𝐴 ∩ Fin)) |
7 | 3, 6 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝑆 ≈ (𝒫 𝐴 ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 ∩ cin 3539 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 –1-1-onto→wf1o 5803 (class class class)co 6549 1𝑜c1o 7440 2𝑜c2o 7441 ↑𝑚 cmap 7744 ≈ cen 7838 Fincfn 7841 finSupp cfsupp 8158 |
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-rep 4699 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-3or 1032 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-reu 2903 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-supp 7183 df-1o 7447 df-2o 7448 df-map 7746 df-en 7842 df-fsupp 8159 |
This theorem is referenced by: frlmpwfi 36686 |
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