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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 2692 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | snex 4117 | . . . . 5 ⊢ {𝑥} ∈ V |
3 | 2 | rnex 4814 | . . . 4 ⊢ ran {𝑥} ∈ V |
4 | 3 | uniex 4367 | . . 3 ⊢ ∪ ran {𝑥} ∈ V |
5 | df-2nd 6047 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
6 | 4, 5 | fnmpti 5259 | . 2 ⊢ 2nd Fn V |
7 | 5 | rnmpt 4795 | . . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
8 | vex 2692 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 8, 8 | opex 4159 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V |
10 | 8, 8 | op2nda 5031 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑦〉} = 𝑦 |
11 | 10 | eqcomi 2144 | . . . . . 6 ⊢ 𝑦 = ∪ ran {〈𝑦, 𝑦〉} |
12 | sneq 3543 | . . . . . . . . . 10 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
13 | 12 | rneqd 4776 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ran {𝑥} = ran {〈𝑦, 𝑦〉}) |
14 | 13 | unieqd 3755 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑦〉}) |
15 | 14 | eqeq2d 2152 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {〈𝑦, 𝑦〉})) |
16 | 15 | rspcev 2793 | . . . . . 6 ⊢ ((〈𝑦, 𝑦〉 ∈ V ∧ 𝑦 = ∪ ran {〈𝑦, 𝑦〉}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
17 | 9, 11, 16 | mp2an 423 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥} |
18 | 8, 17 | 2th 173 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
19 | 18 | abbi2i 2255 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
20 | 7, 19 | eqtr4i 2164 | . 2 ⊢ ran 2nd = V |
21 | df-fo 5137 | . 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V)) | |
22 | 6, 20, 21 | mpbir2an 927 | 1 ⊢ 2nd :V–onto→V |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 {cab 2126 ∃wrex 2418 Vcvv 2689 {csn 3532 〈cop 3535 ∪ cuni 3744 ran crn 4548 Fn wfn 5126 –onto→wfo 5129 2nd c2nd 6045 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-fun 5133 df-fn 5134 df-fo 5137 df-2nd 6047 |
This theorem is referenced by: 2ndcof 6070 2ndexg 6074 df2nd2 6125 2ndconst 6127 suplocexprlemmu 7550 suplocexprlemdisj 7552 suplocexprlemloc 7553 suplocexprlemub 7555 upxp 12480 uptx 12482 cnmpt2nd 12497 |
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