Theorem fo1st 7334

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 5036	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7245	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7099	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7314	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6162	. 2 ⊢ 1st Fn V
6	4	rnmpt 5509	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3352	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5060	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 5760	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2779	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4324	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5464	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4582	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	eqeq2d 2780	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ dom {𝑥} ↔ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}))
15	14	rspcev 3458	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
16	8, 10, 15	mp2an 664	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
17	7, 16	2th 254	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
18	17	abbi2i 2886	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
19	6, 18	eqtr4i 2795	. 2 ⊢ ran 1st = V
20		df-fo 6037	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
21	5, 19, 20	mpbir2an 682	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1630   ∈ wcel 2144  {cab 2756  ∃wrex 3061  Vcvv 3349  {csn 4314  ⟨cop 4320  ∪ cuni 4572  dom cdm 5249  ran crn 5250   Fn wfn 6026  –onto→wfo 6029  1^st c1st 7312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-fun 6033  df-fn 6034  df-fo 6037  df-1st 7314

This theorem is referenced by:  br1steqg  7336  1stcof  7344  df1st2  7413  1stconst  7415  fsplit  7432  algrflem  7436  fpwwe  9669  axpre-sup  10191  homadm  16896  homacd  16897  dmaf  16905  cdaf  16906  1stf1  17039  1stf2  17040  1stfcl  17044  upxp  21646  uptx  21648  cnmpt1st  21691  bcthlem4  23342  uniiccdif  23565  vafval  27792  smfval  27794  0vfval  27795  vsfval  27822  xppreima  29783  xppreima2  29784  1stpreimas  29817  1stpreima  29818  gsummpt2d  30115  cnre2csqima  30291  poimirlem26  33761  poimirlem27  33762