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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 5439 | . . . . 5 ⊢ {𝑥} ∈ V | |
2 | 1 | dmex 7931 | . . . 4 ⊢ dom {𝑥} ∈ V |
3 | 2 | uniex 7759 | . . 3 ⊢ ∪ dom {𝑥} ∈ V |
4 | df-1st 8012 | . . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
5 | 3, 4 | fnmpti 6711 | . 2 ⊢ 1st Fn V |
6 | 4 | rnmpt 5970 | . . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
7 | vex 3481 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | opex 5474 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V | |
9 | 7, 7 | op1sta 6246 | . . . . . . 7 ⊢ ∪ dom {〈𝑦, 𝑦〉} = 𝑦 |
10 | 9 | eqcomi 2743 | . . . . . 6 ⊢ 𝑦 = ∪ dom {〈𝑦, 𝑦〉} |
11 | sneq 4640 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
12 | 11 | dmeqd 5918 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → dom {𝑥} = dom {〈𝑦, 𝑦〉}) |
13 | 12 | unieqd 4924 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ dom {𝑥} = ∪ dom {〈𝑦, 𝑦〉}) |
14 | 13 | rspceeqv 3644 | . . . . . 6 ⊢ ((〈𝑦, 𝑦〉 ∈ V ∧ 𝑦 = ∪ dom {〈𝑦, 𝑦〉}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}) |
15 | 8, 10, 14 | mp2an 692 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥} |
16 | 7, 15 | 2th 264 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}) |
17 | 16 | eqabi 2874 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
18 | 6, 17 | eqtr4i 2765 | . 2 ⊢ ran 1st = V |
19 | df-fo 6568 | . 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V)) | |
20 | 5, 18, 19 | mpbir2an 711 | 1 ⊢ 1st :V–onto→V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 {cab 2711 ∃wrex 3067 Vcvv 3477 {csn 4630 〈cop 4636 ∪ cuni 4911 dom cdm 5688 ran crn 5689 Fn wfn 6557 –onto→wfo 6560 1st c1st 8010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-fun 6564 df-fn 6565 df-fo 6568 df-1st 8012 |
This theorem is referenced by: br1steqg 8034 1stcof 8042 df1st2 8121 1stconst 8123 fsplit 8140 opco1 8146 fpwwe 10683 axpre-sup 11206 homadm 18093 homacd 18094 dmaf 18102 cdaf 18103 1stf1 18247 1stf2 18248 1stfcl 18252 upxp 23646 uptx 23648 cnmpt1st 23691 bcthlem4 25374 uniiccdif 25626 precsexlem10 28254 precsexlem11 28255 vafval 30631 smfval 30633 0vfval 30634 vsfval 30661 xppreima 32661 xppreima2 32667 1stpreimas 32720 1stpreima 32721 fsuppcurry2 32743 gsummpt2d 33034 cnre2csqima 33871 poimirlem26 37632 poimirlem27 37633 |
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