Theorem fo1st 8022

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 5392	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7922	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7759	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 8002	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6683	. 2 ⊢ 1st Fn V
6	4	rnmpt 5938	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5431	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6226	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2769	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4590	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5886	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4879	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3598	. . . . . 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 2895	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2786	. 2 ⊢ ran 1st = V
19		df-fo 6545	. 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 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450  {csn 4584  ⟨cop 4834  ∪ cuni 4866  dom cdm 5650  ran crn 5651   Fn wfn 6534  –onto→wfo 6537  1^st c1st 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-fun 6541  df-fn 6542  df-fo 6545  df-1st 8002

This theorem is used by:  br1steqg  8024  1stcof  8032  df1st2  8112  1stconst  8114  fsplit  8131  opco1  8137  fpwwe  10735  axpre-sup  11258  homadm  18221  homacd  18222  dmaf  18230  cdaf  18231  1stf1  18372  1stf2  18373  1stfcl  18377  upxp  23948  uptx  23950  cnmpt1st  23993  bcthlem4  25654  uniiccdif  25905  precsexlem10  28608  precsexlem11  28609  vafval  31211  smfval  31213  0vfval  31214  vsfval  31241  xppreima  33287  xppreima2  33293  1stpreimas  33347  1stpreima  33348  fsuppcurry2  33364  gsummpt2d  33657  cnre2csqima  34590  poimirlem26  38658  poimirlem27  38659