Theorem fo1st 7951

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 5364	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7849	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7684	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7931	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6628	. 2 ⊢ 1st Fn V
6	4	rnmpt 5899	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3435	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5403	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6176	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2748	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4565	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5847	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4851	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3583	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
15	8, 10, 14	mp2an 698	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
16	7, 15	2th 265	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	eqabi 2874	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2765	. 2 ⊢ ran 1st = V
19		df-fo 6491	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 717	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1547   ∈ wcel 2119  {cab 2717  ∃wrex 3063  Vcvv 3431  {csn 4555  ⟨cop 4561  ∪ cuni 4838  dom cdm 5618  ran crn 5619   Fn wfn 6480  –onto→wfo 6483  1^st c1st 7929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-fun 6487  df-fn 6488  df-fo 6491  df-1st 7931

This theorem is referenced by:  br1steqg  7953  1stcof  7961  df1st2  8037  1stconst  8039  fsplit  8056  opco1  8062  fpwwe  10560  axpre-sup  11083  homadm  17998  homacd  17999  dmaf  18007  cdaf  18008  1stf1  18149  1stf2  18150  1stfcl  18154  upxp  23606  uptx  23608  cnmpt1st  23651  bcthlem4  25312  uniiccdif  25563  precsexlem10  28226  precsexlem11  28227  vafval  30692  smfval  30694  0vfval  30695  vsfval  30722  xppreima  32737  xppreima2  32743  1stpreimas  32798  1stpreima  32799  fsuppcurry2  32817  gsummpt2d  33130  cnre2csqima  34095  poimirlem26  38013  poimirlem27  38014