Theorem fo1st 7703

Description: The 1^st 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.

Step	Hyp	Ref	Expression
1		snex 5323	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7610	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7461	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7683	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6485	. 2 ⊢ 1st Fn V
6	4	rnmpt 5821	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3497	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5348	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6076	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2830	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4570	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5768	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4841	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3637	. . . . . 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 266	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	abbi2i 2953	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2847	. 2 ⊢ ran 1st = V
19		df-fo 6355	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 709	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1533   ∈ wcel 2110  {cab 2799  ∃wrex 3139  Vcvv 3494  {csn 4560  ⟨cop 4566  ∪ cuni 4831  dom cdm 5549  ran crn 5550   Fn wfn 6344  –onto→wfo 6347  1^st c1st 7681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-fun 6351  df-fn 6352  df-fo 6355  df-1st 7683

This theorem is referenced by:  br1steqg  7705  1stcof  7713  df1st2  7787  1stconst  7789  fsplit  7806  fsplitOLD  7807  algrflem  7813  fpwwe  10062  axpre-sup  10585  homadm  17294  homacd  17295  dmaf  17303  cdaf  17304  1stf1  17436  1stf2  17437  1stfcl  17441  upxp  22225  uptx  22227  cnmpt1st  22270  bcthlem4  23924  uniiccdif  24173  vafval  28374  smfval  28376  0vfval  28377  vsfval  28404  xppreima  30388  xppreima2  30389  1stpreimas  30435  1stpreima  30436  fsuppcurry2  30456  gsummpt2d  30682  cnre2csqima  31149  poimirlem26  34912  poimirlem27  34913