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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 2733 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | snex 4171 | . . . . 5 ⊢ {𝑥} ∈ V |
3 | 2 | rnex 4878 | . . . 4 ⊢ ran {𝑥} ∈ V |
4 | 3 | uniex 4422 | . . 3 ⊢ ∪ ran {𝑥} ∈ V |
5 | df-2nd 6120 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
6 | 4, 5 | fnmpti 5326 | . 2 ⊢ 2nd Fn V |
7 | 5 | rnmpt 4859 | . . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
8 | vex 2733 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 8, 8 | opex 4214 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V |
10 | 8, 8 | op2nda 5095 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑦〉} = 𝑦 |
11 | 10 | eqcomi 2174 | . . . . . 6 ⊢ 𝑦 = ∪ ran {〈𝑦, 𝑦〉} |
12 | sneq 3594 | . . . . . . . . . 10 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
13 | 12 | rneqd 4840 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ran {𝑥} = ran {〈𝑦, 𝑦〉}) |
14 | 13 | unieqd 3807 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑦〉}) |
15 | 14 | eqeq2d 2182 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {〈𝑦, 𝑦〉})) |
16 | 15 | rspcev 2834 | . . . . . 6 ⊢ ((〈𝑦, 𝑦〉 ∈ V ∧ 𝑦 = ∪ ran {〈𝑦, 𝑦〉}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
17 | 9, 11, 16 | mp2an 424 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥} |
18 | 8, 17 | 2th 173 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}) |
19 | 18 | abbi2i 2285 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}} |
20 | 7, 19 | eqtr4i 2194 | . 2 ⊢ ran 2nd = V |
21 | df-fo 5204 | . 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V)) | |
22 | 6, 20, 21 | mpbir2an 937 | 1 ⊢ 2nd :V–onto→V |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 {cab 2156 ∃wrex 2449 Vcvv 2730 {csn 3583 〈cop 3586 ∪ cuni 3796 ran crn 4612 Fn wfn 5193 –onto→wfo 5196 2nd c2nd 6118 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-fun 5200 df-fn 5201 df-fo 5204 df-2nd 6120 |
This theorem is referenced by: 2ndcof 6143 2ndexg 6147 df2nd2 6199 2ndconst 6201 suplocexprlemmu 7680 suplocexprlemdisj 7682 suplocexprlemloc 7683 suplocexprlemub 7685 upxp 13066 uptx 13068 cnmpt2nd 13083 |
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