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Mirrors > Home > ILE Home > Th. List > fo1st | GIF version |
Description: The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
fo1st | ⊢ 1st :V–onto→V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2715 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | snex 4146 | . . . . 5 ⊢ {𝑥} ∈ V |
3 | 2 | dmex 4852 | . . . 4 ⊢ dom {𝑥} ∈ V |
4 | 3 | uniex 4397 | . . 3 ⊢ ∪ dom {𝑥} ∈ V |
5 | df-1st 6088 | . . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
6 | 4, 5 | fnmpti 5298 | . 2 ⊢ 1st Fn V |
7 | 5 | rnmpt 4834 | . . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
8 | vex 2715 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 8, 8 | opex 4189 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V |
10 | 8, 8 | op1sta 5067 | . . . . . . 7 ⊢ ∪ dom {〈𝑦, 𝑦〉} = 𝑦 |
11 | 10 | eqcomi 2161 | . . . . . 6 ⊢ 𝑦 = ∪ dom {〈𝑦, 𝑦〉} |
12 | sneq 3571 | . . . . . . . . . 10 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
13 | 12 | dmeqd 4788 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → dom {𝑥} = dom {〈𝑦, 𝑦〉}) |
14 | 13 | unieqd 3783 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ dom {𝑥} = ∪ dom {〈𝑦, 𝑦〉}) |
15 | 14 | eqeq2d 2169 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → (𝑦 = ∪ dom {𝑥} ↔ 𝑦 = ∪ dom {〈𝑦, 𝑦〉})) |
16 | 15 | rspcev 2816 | . . . . . 6 ⊢ ((〈𝑦, 𝑦〉 ∈ V ∧ 𝑦 = ∪ dom {〈𝑦, 𝑦〉}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}) |
17 | 9, 11, 16 | mp2an 423 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥} |
18 | 8, 17 | 2th 173 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}) |
19 | 18 | abbi2i 2272 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
20 | 7, 19 | eqtr4i 2181 | . 2 ⊢ ran 1st = V |
21 | df-fo 5176 | . 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V)) | |
22 | 6, 20, 21 | mpbir2an 927 | 1 ⊢ 1st :V–onto→V |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 {cab 2143 ∃wrex 2436 Vcvv 2712 {csn 3560 〈cop 3563 ∪ cuni 3772 dom cdm 4586 ran crn 4587 Fn wfn 5165 –onto→wfo 5168 1st c1st 6086 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-fun 5172 df-fn 5173 df-fo 5176 df-1st 6088 |
This theorem is referenced by: 1stcof 6111 1stexg 6115 df1st2 6166 1stconst 6168 algrflem 6176 algrflemg 6177 suplocexprlemell 7633 suplocexprlem2b 7634 suplocexprlemlub 7644 upxp 12672 uptx 12674 cnmpt1st 12688 |
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