Theorem fo2nd 7996

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 5430	. . . . 5 ⊢ {𝑥} ∈ V
2	1	rnex 7903	. . . 4 ⊢ ran {𝑥} ∈ V
3	2	uniex 7731	. . 3 ⊢ ∪ ran {𝑥} ∈ V
4		df-2nd 7976	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5	3, 4	fnmpti 6694	. 2 ⊢ 2nd Fn V
6	4	rnmpt 5955	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
7		vex 3479	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5465	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op2nda 6228	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2742	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
11		sneq 4639	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	rneqd 5938	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
13	12	unieqd 4923	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3634	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
15	8, 10, 14	mp2an 691	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}
16	7, 15	2th 264	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
17	16	eqabi 2870	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
18	6, 17	eqtr4i 2764	. 2 ⊢ ran 2nd = V
19		df-fo 6550	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
20	5, 18, 19	mpbir2an 710	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071  Vcvv 3475  {csn 4629  ⟨cop 4635  ∪ cuni 4909  ran crn 5678   Fn wfn 6539  –onto→wfo 6542  2^nd c2nd 7974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-fo 6550  df-2nd 7976

This theorem is referenced by:  br2ndeqg  7998  2ndcof  8006  df2nd2  8085  2ndconst  8087  opco2  8110  iunfo  10534  cdaf  18000  2ndf1  18147  2ndf2  18148  2ndfcl  18150  gsum2dlem2  19839  upxp  23127  uptx  23129  cnmpt2nd  23173  uniiccdif  25095  precsexlem10  27662  precsexlem11  27663  xppreima  31871  2ndimaxp  31872  2ndresdju  31874  xppreima2  31876  2ndpreima  31929  fsuppcurry1  31950  gsummpt2d  32201  gsumpart  32207  cnre2csqima  32891  filnetlem4  35266