Theorem fo1st 8034

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 5434	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7931	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7761	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 8014	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6711	. 2 ⊢ 1st Fn V
6	4	rnmpt 5968	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3484	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5469	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6245	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2746	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4636	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5916	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4920	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3645	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
15	8, 10, 14	mp2an 692	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
16	7, 15	2th 264	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	eqabi 2877	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2768	. 2 ⊢ ran 1st = V
19		df-fo 6567	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 711	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070  Vcvv 3480  {csn 4626  ⟨cop 4632  ∪ cuni 4907  dom cdm 5685  ran crn 5686   Fn wfn 6556  –onto→wfo 6559  1^st c1st 8012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-fo 6567  df-1st 8014

This theorem is referenced by:  br1steqg  8036  1stcof  8044  df1st2  8123  1stconst  8125  fsplit  8142  opco1  8148  fpwwe  10686  axpre-sup  11209  homadm  18085  homacd  18086  dmaf  18094  cdaf  18095  1stf1  18237  1stf2  18238  1stfcl  18242  upxp  23631  uptx  23633  cnmpt1st  23676  bcthlem4  25361  uniiccdif  25613  precsexlem10  28240  precsexlem11  28241  vafval  30622  smfval  30624  0vfval  30625  vsfval  30652  xppreima  32655  xppreima2  32661  1stpreimas  32715  1stpreima  32716  fsuppcurry2  32737  gsummpt2d  33052  cnre2csqima  33910  poimirlem26  37653  poimirlem27  37654