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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 | snex 4835 | . . . . 5 ⊢ {𝑥} ∈ V | |
2 | 1 | dmex 6991 | . . . 4 ⊢ dom {𝑥} ∈ V |
3 | 2 | uniex 6851 | . . 3 ⊢ ∪ dom {𝑥} ∈ V |
4 | df-1st 7059 | . . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
5 | 3, 4 | fnmpti 5935 | . 2 ⊢ 1st Fn V |
6 | 4 | rnmpt 5292 | . . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
7 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | opex 4859 | . . . . . 6 ⊢ 〈𝑦, 𝑦〉 ∈ V | |
9 | 7, 7 | op1sta 5535 | . . . . . . 7 ⊢ ∪ dom {〈𝑦, 𝑦〉} = 𝑦 |
10 | 9 | eqcomi 2619 | . . . . . 6 ⊢ 𝑦 = ∪ dom {〈𝑦, 𝑦〉} |
11 | sneq 4135 | . . . . . . . . . 10 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → {𝑥} = {〈𝑦, 𝑦〉}) | |
12 | 11 | dmeqd 5248 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → dom {𝑥} = dom {〈𝑦, 𝑦〉}) |
13 | 12 | unieqd 4382 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → ∪ dom {𝑥} = ∪ dom {〈𝑦, 𝑦〉}) |
14 | 13 | eqeq2d 2620 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑦〉 → (𝑦 = ∪ dom {𝑥} ↔ 𝑦 = ∪ dom {〈𝑦, 𝑦〉})) |
15 | 14 | rspcev 3282 | . . . . . 6 ⊢ ((〈𝑦, 𝑦〉 ∈ V ∧ 𝑦 = ∪ dom {〈𝑦, 𝑦〉}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}) |
16 | 8, 10, 15 | mp2an 704 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥} |
17 | 7, 16 | 2th 253 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}) |
18 | 17 | abbi2i 2725 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}} |
19 | 6, 18 | eqtr4i 2635 | . 2 ⊢ ran 1st = V |
20 | df-fo 5810 | . 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V)) | |
21 | 5, 19, 20 | mpbir2an 957 | 1 ⊢ 1st :V–onto→V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 Vcvv 3173 {csn 4125 〈cop 4131 ∪ cuni 4372 dom cdm 5038 ran crn 5039 Fn wfn 5799 –onto→wfo 5802 1st c1st 7057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-fo 5810 df-1st 7059 |
This theorem is referenced by: 1stcof 7087 df1st2 7150 1stconst 7152 fsplit 7169 algrflem 7173 fpwwe 9347 axpre-sup 9869 homadm 16513 homacd 16514 dmaf 16522 cdaf 16523 1stf1 16655 1stf2 16656 1stfcl 16660 upxp 21236 uptx 21238 cnmpt1st 21281 bcthlem4 22932 uniiccdif 23152 vafval 26842 smfval 26844 0vfval 26845 vsfval 26872 xppreima 28829 xppreima2 28830 1stpreimas 28866 1stpreima 28867 gsummpt2d 29112 cnre2csqima 29285 br1steq 30917 poimirlem26 32605 poimirlem27 32606 |
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