![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fo1st | Structured version Visualization version 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 | vsnex 5434 | . . . . 5 ⊢ {𝑥} ∈ V | |
2 | 1 | dmex 7921 | . . . 4 ⊢ dom {𝑥} ∈ V |
3 | 2 | uniex 7751 | . . 3 ⊢ ∪ dom {𝑥} ∈ V |
4 | df-1st 8002 | . . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
5 | 3, 4 | fnmpti 6703 | . 2 ⊢ 1st Fn V |
6 | 4 | rnmpt 5960 | . . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
7 | vex 3465 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | opex 5469 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V | |
9 | 7, 7 | op1sta 6235 | . . . . . . 7 ⊢ ∪ dom {〈𝑦, 𝑦〉} = 𝑦 |
10 | 9 | eqcomi 2734 | . . . . . 6 ⊢ 𝑦 = ∪ dom {〈𝑦, 𝑦〉} |
11 | sneq 4642 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
12 | 11 | dmeqd 5911 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → dom {𝑥} = dom {〈𝑦, 𝑦〉}) |
13 | 12 | unieqd 4925 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ dom {𝑥} = ∪ dom {〈𝑦, 𝑦〉}) |
14 | 13 | rspceeqv 3629 | . . . . . 6 ⊢ ((〈𝑦, 𝑦〉 ∈ V ∧ 𝑦 = ∪ dom {〈𝑦, 𝑦〉}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}) |
15 | 8, 10, 14 | mp2an 690 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥} |
16 | 7, 15 | 2th 263 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}) |
17 | 16 | eqabi 2861 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
18 | 6, 17 | eqtr4i 2756 | . 2 ⊢ ran 1st = V |
19 | df-fo 6559 | . 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V)) | |
20 | 5, 18, 19 | mpbir2an 709 | 1 ⊢ 1st :V–onto→V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cab 2702 ∃wrex 3059 Vcvv 3461 {csn 4632 〈cop 4638 ∪ cuni 4912 dom cdm 5681 ran crn 5682 Fn wfn 6548 –onto→wfo 6551 1st c1st 8000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-fun 6555 df-fn 6556 df-fo 6559 df-1st 8002 |
This theorem is referenced by: br1steqg 8024 1stcof 8032 df1st2 8111 1stconst 8113 fsplit 8130 opco1 8136 fpwwe 10685 axpre-sup 11208 homadm 18057 homacd 18058 dmaf 18066 cdaf 18067 1stf1 18211 1stf2 18212 1stfcl 18216 upxp 23610 uptx 23612 cnmpt1st 23655 bcthlem4 25338 uniiccdif 25590 precsexlem10 28207 precsexlem11 28208 vafval 30528 smfval 30530 0vfval 30531 vsfval 30558 xppreima 32554 xppreima2 32559 1stpreimas 32608 1stpreima 32609 fsuppcurry2 32631 gsummpt2d 32895 cnre2csqima 33682 poimirlem26 37295 poimirlem27 37296 |
Copyright terms: Public domain | W3C validator |