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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 5439 | . . . . 5 ⊢ {𝑥} ∈ V | |
2 | 1 | rnex 7932 | . . . 4 ⊢ ran {𝑥} ∈ V |
3 | 2 | uniex 7759 | . . 3 ⊢ ∪ ran {𝑥} ∈ V |
4 | df-2nd 8013 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
5 | 3, 4 | fnmpti 6711 | . 2 ⊢ 2nd Fn V |
6 | 4 | rnmpt 5970 | . . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
7 | vex 3481 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | opex 5474 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V | |
9 | 7, 7 | op2nda 6249 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑦〉} = 𝑦 |
10 | 9 | eqcomi 2743 | . . . . . 6 ⊢ 𝑦 = ∪ ran {〈𝑦, 𝑦〉} |
11 | sneq 4640 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
12 | 11 | rneqd 5951 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ran {𝑥} = ran {〈𝑦, 𝑦〉}) |
13 | 12 | unieqd 4924 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑦〉}) |
14 | 13 | rspceeqv 3644 | . . . . . 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 2874 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
18 | 6, 17 | eqtr4i 2765 | . 2 ⊢ ran 2nd = V |
19 | df-fo 6568 | . 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 1536 ∈ wcel 2105 {cab 2711 ∃wrex 3067 Vcvv 3477 {csn 4630 〈cop 4636 ∪ cuni 4911 ran crn 5689 Fn wfn 6557 –onto→wfo 6560 2nd c2nd 8011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-fun 6564 df-fn 6565 df-fo 6568 df-2nd 8013 |
This theorem is referenced by: br2ndeqg 8035 2ndcof 8043 df2nd2 8122 2ndconst 8124 opco2 8147 iunfo 10576 cdaf 18103 2ndf1 18250 2ndf2 18251 2ndfcl 18253 gsum2dlem2 20003 upxp 23646 uptx 23648 cnmpt2nd 23692 uniiccdif 25626 precsexlem10 28254 precsexlem11 28255 xppreima 32661 2ndimaxp 32662 2ndresdju 32665 xppreima2 32667 2ndpreima 32722 fsuppcurry1 32742 gsummpt2d 33034 gsumpart 33042 cnre2csqima 33871 filnetlem4 36363 |
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