Theorem fo1st 8002

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		vsnex 5404	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7902	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7736	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7982	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6676	. 2 ⊢ 1st Fn V
6	4	rnmpt 5945	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3467	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5443	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6224	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2778	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4601	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5893	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4886	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3613	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
15	8, 10, 14	mp2an 704	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
16	7, 15	2th 267	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	eqabi 2904	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2795	. 2 ⊢ ran 1st = V
19		df-fo 6540	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 723	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095  Vcvv 3463  {csn 4591  ⟨cop 4597  ∪ cuni 4873  dom cdm 5659  ran crn 5660   Fn wfn 6529  –onto→wfo 6532  1^st c1st 7980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6536  df-fn 6537  df-fo 6540  df-1st 7982

This theorem is referenced by:  br1steqg  8004  1stcof  8012  df1st2  8089  1stconst  8091  fsplit  8108  opco1  8114  fpwwe  10627  axpre-sup  11150  homadm  18093  homacd  18094  dmaf  18102  cdaf  18103  1stf1  18244  1stf2  18245  1stfcl  18249  upxp  23745  uptx  23747  cnmpt1st  23790  bcthlem4  25451  uniiccdif  25702  precsexlem10  28371  precsexlem11  28372  vafval  30892  smfval  30894  0vfval  30895  vsfval  30922  xppreima  32927  xppreima2  32933  1stpreimas  32988  1stpreima  32989  fsuppcurry2  33007  gsummpt2d  33306  cnre2csqima  34242  poimirlem26  38180  poimirlem27  38181