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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 5332 | . . . . 5 ⊢ {𝑥} ∈ V | |
2 | 1 | rnex 7617 | . . . 4 ⊢ ran {𝑥} ∈ V |
3 | 2 | uniex 7467 | . . 3 ⊢ ∪ ran {𝑥} ∈ V |
4 | df-2nd 7690 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
5 | 3, 4 | fnmpti 6491 | . 2 ⊢ 2nd Fn V |
6 | 4 | rnmpt 5827 | . . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
7 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | opex 5356 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V | |
9 | 7, 7 | op2nda 6085 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑦〉} = 𝑦 |
10 | 9 | eqcomi 2830 | . . . . . 6 ⊢ 𝑦 = ∪ ran {〈𝑦, 𝑦〉} |
11 | sneq 4577 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
12 | 11 | rneqd 5808 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ran {𝑥} = ran {〈𝑦, 𝑦〉}) |
13 | 12 | unieqd 4852 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑦〉}) |
14 | 13 | rspceeqv 3638 | . . . . . 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 2953 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
18 | 6, 17 | eqtr4i 2847 | . 2 ⊢ ran 2nd = V |
19 | df-fo 6361 | . 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 2799 ∃wrex 3139 Vcvv 3494 {csn 4567 〈cop 4573 ∪ cuni 4838 ran crn 5556 Fn wfn 6350 –onto→wfo 6353 2nd c2nd 7688 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-fun 6357 df-fn 6358 df-fo 6361 df-2nd 7690 |
This theorem is referenced by: br2ndeqg 7712 2ndcof 7720 df2nd2 7794 2ndconst 7796 iunfo 9961 cdaf 17310 2ndf1 17445 2ndf2 17446 2ndfcl 17448 gsum2dlem2 19091 upxp 22231 uptx 22233 cnmpt2nd 22277 uniiccdif 24179 xppreima 30394 xppreima2 30395 2ndpreima 30443 fsuppcurry1 30461 gsummpt2d 30687 cnre2csqima 31154 filnetlem4 33729 |
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