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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 | snex 5334 | . . . . 5 ⊢ {𝑥} ∈ V | |
2 | 1 | rnex 7619 | . . . 4 ⊢ ran {𝑥} ∈ V |
3 | 2 | uniex 7469 | . . 3 ⊢ ∪ ran {𝑥} ∈ V |
4 | df-2nd 7692 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
5 | 3, 4 | fnmpti 6493 | . 2 ⊢ 2nd Fn V |
6 | 4 | rnmpt 5829 | . . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
7 | vex 3499 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | opex 5358 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V | |
9 | 7, 7 | op2nda 6087 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑦〉} = 𝑦 |
10 | 9 | eqcomi 2832 | . . . . . 6 ⊢ 𝑦 = ∪ ran {〈𝑦, 𝑦〉} |
11 | sneq 4579 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
12 | 11 | rneqd 5810 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ran {𝑥} = ran {〈𝑦, 𝑦〉}) |
13 | 12 | unieqd 4854 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑦〉}) |
14 | 13 | rspceeqv 3640 | . . . . . 6 ⊢ ((〈𝑦, 𝑦〉 ∈ V ∧ 𝑦 = ∪ ran {〈𝑦, 𝑦〉}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
15 | 8, 10, 14 | mp2an 690 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥} |
16 | 7, 15 | 2th 266 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
17 | 16 | abbi2i 2955 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
18 | 6, 17 | eqtr4i 2849 | . 2 ⊢ ran 2nd = V |
19 | df-fo 6363 | . 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V)) | |
20 | 5, 18, 19 | mpbir2an 709 | 1 ⊢ 2nd :V–onto→V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {cab 2801 ∃wrex 3141 Vcvv 3496 {csn 4569 〈cop 4575 ∪ cuni 4840 ran crn 5558 Fn wfn 6352 –onto→wfo 6355 2nd c2nd 7690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-fo 6363 df-2nd 7692 |
This theorem is referenced by: br2ndeqg 7714 2ndcof 7722 df2nd2 7796 2ndconst 7798 iunfo 9963 cdaf 17312 2ndf1 17447 2ndf2 17448 2ndfcl 17450 gsum2dlem2 19093 upxp 22233 uptx 22235 cnmpt2nd 22279 uniiccdif 24181 xppreima 30396 xppreima2 30397 2ndpreima 30445 fsuppcurry1 30463 gsummpt2d 30689 cnre2csqima 31156 filnetlem4 33731 |
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