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Theorem fo1st 7695
Description: The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fo1st 1st :V–onto→V

Proof of Theorem fo1st
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 5300 . . . . 5 {𝑥} ∈ V
21dmex 7602 . . . 4 dom {𝑥} ∈ V
32uniex 7451 . . 3 dom {𝑥} ∈ V
4 df-1st 7675 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
53, 4fnmpti 6467 . 2 1st Fn V
64rnmpt 5795 . . 3 ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
7 vex 3447 . . . . 5 𝑦 ∈ V
8 opex 5324 . . . . . 6 𝑦, 𝑦⟩ ∈ V
97, 7op1sta 6053 . . . . . . 7 dom {⟨𝑦, 𝑦⟩} = 𝑦
109eqcomi 2810 . . . . . 6 𝑦 = dom {⟨𝑦, 𝑦⟩}
11 sneq 4538 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1211dmeqd 5742 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1312unieqd 4817 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1413rspceeqv 3589 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = dom {𝑥})
158, 10, 14mp2an 691 . . . . 5 𝑥 ∈ V 𝑦 = dom {𝑥}
167, 152th 267 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = dom {𝑥})
1716abbi2i 2932 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
186, 17eqtr4i 2827 . 2 ran 1st = V
19 df-fo 6334 . 2 (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
205, 18, 19mpbir2an 710 1 1st :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2112  {cab 2779  wrex 3110  Vcvv 3444  {csn 4528  cop 4534   cuni 4803  dom cdm 5523  ran crn 5524   Fn wfn 6323  ontowfo 6326  1st c1st 7673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-fun 6330  df-fn 6331  df-fo 6334  df-1st 7675
This theorem is referenced by:  br1steqg  7697  1stcof  7705  df1st2  7780  1stconst  7782  fsplit  7799  fsplitOLD  7800  algrflem  7806  fpwwe  10061  axpre-sup  10584  homadm  17295  homacd  17296  dmaf  17304  cdaf  17305  1stf1  17437  1stf2  17438  1stfcl  17442  upxp  22231  uptx  22233  cnmpt1st  22276  bcthlem4  23934  uniiccdif  24185  vafval  28389  smfval  28391  0vfval  28392  vsfval  28419  xppreima  30411  xppreima2  30416  1stpreimas  30468  1stpreima  30469  fsuppcurry2  30491  gsummpt2d  30737  cnre2csqima  31262  poimirlem26  35076  poimirlem27  35077
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