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Mirrors > Home > ILE Home > Th. List > fo2nd | 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 | vex 2554 | . . . . . 6 ⊢ x ∈ V | |
2 | snexgOLD 3926 | . . . . . 6 ⊢ (x ∈ V → {x} ∈ V) | |
3 | 1, 2 | ax-mp 7 | . . . . 5 ⊢ {x} ∈ V |
4 | 3 | rnex 4542 | . . . 4 ⊢ ran {x} ∈ V |
5 | 4 | uniex 4140 | . . 3 ⊢ ∪ ran {x} ∈ V |
6 | df-2nd 5710 | . . 3 ⊢ 2nd = (x ∈ V ↦ ∪ ran {x}) | |
7 | 5, 6 | fnmpti 4970 | . 2 ⊢ 2nd Fn V |
8 | 6 | rnmpt 4525 | . . 3 ⊢ ran 2nd = {y ∣ ∃x ∈ V y = ∪ ran {x}} |
9 | vex 2554 | . . . . 5 ⊢ y ∈ V | |
10 | 9, 9 | opex 3957 | . . . . . 6 ⊢ 〈y, y〉 ∈ V |
11 | 9, 9 | op2nda 4748 | . . . . . . 7 ⊢ ∪ ran {〈y, y〉} = y |
12 | 11 | eqcomi 2041 | . . . . . 6 ⊢ y = ∪ ran {〈y, y〉} |
13 | sneq 3378 | . . . . . . . . . 10 ⊢ (x = 〈y, y〉 → {x} = {〈y, y〉}) | |
14 | 13 | rneqd 4506 | . . . . . . . . 9 ⊢ (x = 〈y, y〉 → ran {x} = ran {〈y, y〉}) |
15 | 14 | unieqd 3582 | . . . . . . . 8 ⊢ (x = 〈y, y〉 → ∪ ran {x} = ∪ ran {〈y, y〉}) |
16 | 15 | eqeq2d 2048 | . . . . . . 7 ⊢ (x = 〈y, y〉 → (y = ∪ ran {x} ↔ y = ∪ ran {〈y, y〉})) |
17 | 16 | rspcev 2650 | . . . . . 6 ⊢ ((〈y, y〉 ∈ V ∧ y = ∪ ran {〈y, y〉}) → ∃x ∈ V y = ∪ ran {x}) |
18 | 10, 12, 17 | mp2an 402 | . . . . 5 ⊢ ∃x ∈ V y = ∪ ran {x} |
19 | 9, 18 | 2th 163 | . . . 4 ⊢ (y ∈ V ↔ ∃x ∈ V y = ∪ ran {x}) |
20 | 19 | abbi2i 2149 | . . 3 ⊢ V = {y ∣ ∃x ∈ V y = ∪ ran {x}} |
21 | 8, 20 | eqtr4i 2060 | . 2 ⊢ ran 2nd = V |
22 | df-fo 4851 | . 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V)) | |
23 | 7, 21, 22 | mpbir2an 848 | 1 ⊢ 2nd :V–onto→V |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∈ wcel 1390 {cab 2023 ∃wrex 2301 Vcvv 2551 {csn 3367 〈cop 3370 ∪ cuni 3571 ran crn 4289 Fn wfn 4840 –onto→wfo 4843 2nd c2nd 5708 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-fo 4851 df-2nd 5710 |
This theorem is referenced by: 2ndcof 5733 2ndexg 5737 df2nd2 5783 2ndconst 5785 |
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