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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 2689 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | snex 4109 | . . . . 5 ⊢ {𝑥} ∈ V |
3 | 2 | rnex 4806 | . . . 4 ⊢ ran {𝑥} ∈ V |
4 | 3 | uniex 4359 | . . 3 ⊢ ∪ ran {𝑥} ∈ V |
5 | df-2nd 6039 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
6 | 4, 5 | fnmpti 5251 | . 2 ⊢ 2nd Fn V |
7 | 5 | rnmpt 4787 | . . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
8 | vex 2689 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 8, 8 | opex 4151 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V |
10 | 8, 8 | op2nda 5023 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑦〉} = 𝑦 |
11 | 10 | eqcomi 2143 | . . . . . 6 ⊢ 𝑦 = ∪ ran {〈𝑦, 𝑦〉} |
12 | sneq 3538 | . . . . . . . . . 10 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
13 | 12 | rneqd 4768 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ran {𝑥} = ran {〈𝑦, 𝑦〉}) |
14 | 13 | unieqd 3747 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑦〉}) |
15 | 14 | eqeq2d 2151 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {〈𝑦, 𝑦〉})) |
16 | 15 | rspcev 2789 | . . . . . 6 ⊢ ((〈𝑦, 𝑦〉 ∈ V ∧ 𝑦 = ∪ ran {〈𝑦, 𝑦〉}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
17 | 9, 11, 16 | mp2an 422 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥} |
18 | 8, 17 | 2th 173 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
19 | 18 | abbi2i 2254 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
20 | 7, 19 | eqtr4i 2163 | . 2 ⊢ ran 2nd = V |
21 | df-fo 5129 | . 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V)) | |
22 | 6, 20, 21 | mpbir2an 926 | 1 ⊢ 2nd :V–onto→V |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 {cab 2125 ∃wrex 2417 Vcvv 2686 {csn 3527 〈cop 3530 ∪ cuni 3736 ran crn 4540 Fn wfn 5118 –onto→wfo 5121 2nd c2nd 6037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-fo 5129 df-2nd 6039 |
This theorem is referenced by: 2ndcof 6062 2ndexg 6066 df2nd2 6117 2ndconst 6119 suplocexprlemmu 7526 suplocexprlemdisj 7528 suplocexprlemloc 7529 suplocexprlemub 7531 upxp 12441 uptx 12443 cnmpt2nd 12458 |
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