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| Mirrors > Home > MPE Home > Th. List > fo2nd | Structured version Visualization version GIF version | ||
| Description: The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
| Ref | Expression |
|---|---|
| fo2nd | ⊢ 2nd :V–onto→V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vsnex 5434 | . . . . 5 ⊢ {𝑥} ∈ V | |
| 2 | 1 | rnex 7932 | . . . 4 ⊢ ran {𝑥} ∈ V |
| 3 | 2 | uniex 7761 | . . 3 ⊢ ∪ ran {𝑥} ∈ V |
| 4 | df-2nd 8015 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
| 5 | 3, 4 | fnmpti 6711 | . 2 ⊢ 2nd Fn V |
| 6 | 4 | rnmpt 5968 | . . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
| 7 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 8 | opex 5469 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V | |
| 9 | 7, 7 | op2nda 6248 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑦〉} = 𝑦 |
| 10 | 9 | eqcomi 2746 | . . . . . 6 ⊢ 𝑦 = ∪ ran {〈𝑦, 𝑦〉} |
| 11 | sneq 4636 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
| 12 | 11 | rneqd 5949 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ran {𝑥} = ran {〈𝑦, 𝑦〉}) |
| 13 | 12 | unieqd 4920 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑦〉}) |
| 14 | 13 | rspceeqv 3645 | . . . . . 6 ⊢ ((〈𝑦, 𝑦〉 ∈ V ∧ 𝑦 = ∪ ran {〈𝑦, 𝑦〉}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
| 15 | 8, 10, 14 | mp2an 692 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥} |
| 16 | 7, 15 | 2th 264 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
| 17 | 16 | eqabi 2877 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
| 18 | 6, 17 | eqtr4i 2768 | . 2 ⊢ ran 2nd = V |
| 19 | df-fo 6567 | . 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V)) | |
| 20 | 5, 18, 19 | mpbir2an 711 | 1 ⊢ 2nd :V–onto→V |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 Vcvv 3480 {csn 4626 〈cop 4632 ∪ cuni 4907 ran crn 5686 Fn wfn 6556 –onto→wfo 6559 2nd c2nd 8013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-fo 6567 df-2nd 8015 |
| This theorem is referenced by: br2ndeqg 8037 2ndcof 8045 df2nd2 8124 2ndconst 8126 opco2 8149 iunfo 10579 cdaf 18095 2ndf1 18240 2ndf2 18241 2ndfcl 18243 gsum2dlem2 19989 upxp 23631 uptx 23633 cnmpt2nd 23677 uniiccdif 25613 precsexlem10 28240 precsexlem11 28241 xppreima 32655 2ndimaxp 32656 2ndresdju 32659 xppreima2 32661 2ndpreima 32717 fsuppcurry1 32736 gsummpt2d 33052 gsumpart 33060 cnre2csqima 33910 filnetlem4 36382 |
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