| 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 7931 | . . . 4 ⊢ dom {𝑥} ∈ V |
| 3 | 2 | uniex 7761 | . . 3 ⊢ ∪ dom {𝑥} ∈ V |
| 4 | df-1st 8014 | . . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
| 5 | 3, 4 | fnmpti 6711 | . 2 ⊢ 1st Fn V |
| 6 | 4 | rnmpt 5968 | . . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
| 7 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 8 | opex 5469 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V | |
| 9 | 7, 7 | op1sta 6245 | . . . . . . 7 ⊢ ∪ dom {〈𝑦, 𝑦〉} = 𝑦 |
| 10 | 9 | eqcomi 2746 | . . . . . 6 ⊢ 𝑦 = ∪ dom {〈𝑦, 𝑦〉} |
| 11 | sneq 4636 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
| 12 | 11 | dmeqd 5916 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → dom {𝑥} = dom {〈𝑦, 𝑦〉}) |
| 13 | 12 | unieqd 4920 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ dom {𝑥} = ∪ dom {〈𝑦, 𝑦〉}) |
| 14 | 13 | rspceeqv 3645 | . . . . . 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 2877 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
| 18 | 6, 17 | eqtr4i 2768 | . 2 ⊢ ran 1st = V |
| 19 | df-fo 6567 | . 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 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 Vcvv 3480 {csn 4626 〈cop 4632 ∪ cuni 4907 dom cdm 5685 ran crn 5686 Fn wfn 6556 –onto→wfo 6559 1st c1st 8012 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-fo 6567 df-1st 8014 |
| This theorem is referenced by: br1steqg 8036 1stcof 8044 df1st2 8123 1stconst 8125 fsplit 8142 opco1 8148 fpwwe 10686 axpre-sup 11209 homadm 18085 homacd 18086 dmaf 18094 cdaf 18095 1stf1 18237 1stf2 18238 1stfcl 18242 upxp 23631 uptx 23633 cnmpt1st 23676 bcthlem4 25361 uniiccdif 25613 precsexlem10 28240 precsexlem11 28241 vafval 30622 smfval 30624 0vfval 30625 vsfval 30652 xppreima 32655 xppreima2 32661 1stpreimas 32715 1stpreima 32716 fsuppcurry2 32737 gsummpt2d 33052 cnre2csqima 33910 poimirlem26 37653 poimirlem27 37654 |
| Copyright terms: Public domain | W3C validator |