Theorem fo1st 8008

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 5408	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7908	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7745	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7988	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6682	. 2 ⊢ 1st Fn V
6	4	rnmpt 5949	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3461	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5447	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6228	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2774	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4601	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5897	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4887	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3606	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
15	8, 10, 14	mp2an 705	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
16	7, 15	2th 267	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	eqabi 2900	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2791	. 2 ⊢ ran 1st = V
19		df-fo 6546	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 724	1 ⊢ 1st :V–onto→V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {cab 2743  ∃wrex 3091  Vcvv 3457  {csn 4591  ⟨cop 4597  ∪ cuni 4874  dom cdm 5663  ran crn 5664   Fn wfn 6535  –onto→wfo 6538  1^st c1st 7986

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-fun 6542  df-fn 6543  df-fo 6546  df-1st 7988

This theorem is used by:  br1steqg  8010  1stcof  8018  df1st2  8095  1stconst  8097  fsplit  8114  opco1  8120  fpwwe  10642  axpre-sup  11165  homadm  18115  homacd  18116  dmaf  18124  cdaf  18125  1stf1  18266  1stf2  18267  1stfcl  18271  upxp  23811  uptx  23813  cnmpt1st  23856  bcthlem4  25517  uniiccdif  25768  precsexlem10  28440  precsexlem11  28441  vafval  31002  smfval  31004  0vfval  31005  vsfval  31032  xppreima  33037  xppreima2  33043  1stpreimas  33098  1stpreima  33099  fsuppcurry2  33116  gsummpt2d  33409  cnre2csqima  34341  poimirlem26  38330  poimirlem27  38331