Theorem fo2nd 8016

Description: The 2^nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
fo2nd	⊢ 2nd :V–onto→V

Proof of Theorem fo2nd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vsnex 5411	. . . . 5 ⊢ {𝑥} ∈ V
2	1	rnex 7916	. . . 4 ⊢ ran {𝑥} ∈ V
3	2	uniex 7752	. . 3 ⊢ ∪ ran {𝑥} ∈ V
4		df-2nd 7996	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5	3, 4	fnmpti 6685	. 2 ⊢ 2nd Fn V
6	4	rnmpt 5952	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
7		vex 3462	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5450	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op2nda 6234	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2775	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
11		sneq 4604	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	rneqd 5933	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
13	12	unieqd 4890	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3607	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
15	8, 10, 14	mp2an 705	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}
16	7, 15	2th 267	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
17	16	eqabi 2901	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
18	6, 17	eqtr4i 2792	. 2 ⊢ ran 2nd = V
19		df-fo 6549	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
20	5, 18, 19	mpbir2an 724	1 ⊢ 2nd :V–onto→V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {cab 2744  ∃wrex 3092  Vcvv 3458  {csn 4594  ⟨cop 4600  ∪ cuni 4877  ran crn 5667   Fn wfn 6538  –onto→wfo 6541  2^nd c2nd 7994

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 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-fun 6545  df-fn 6546  df-fo 6549  df-2nd 7996

This theorem is used by:  br2ndeqg  8018  2ndcof  8026  df2nd2  8103  2ndconst  8105  opco2  8128  iunfo  10541  cdaf  18132  2ndf1  18276  2ndf2  18277  2ndfcl  18279  gsum2dlem2  20072  upxp  23817  uptx  23819  cnmpt2nd  23863  uniiccdif  25774  precsexlem10  28446  precsexlem11  28447  xppreima  33027  2ndimaxp  33028  2ndresdju  33031  xppreima2  33033  2ndpreima  33090  fsuppcurry1  33106  gsummpt2d  33400  gsumpart  33414  cnre2csqima  34332  filnetlem4  36932