Theorem fo2nd 7952

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 5376	. . . . 5 ⊢ {𝑥} ∈ V
2	1	rnex 7850	. . . 4 ⊢ ran {𝑥} ∈ V
3	2	uniex 7681	. . 3 ⊢ ∪ ran {𝑥} ∈ V
4		df-2nd 7932	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5	3, 4	fnmpti 6629	. 2 ⊢ 2nd Fn V
6	4	rnmpt 5903	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
7		vex 3442	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5411	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op2nda 6181	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2738	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
11		sneq 4589	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	rneqd 5884	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
13	12	unieqd 4874	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3602	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
15	8, 10, 14	mp2an 692	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}
16	7, 15	2th 264	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
17	16	eqabi 2863	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
18	6, 17	eqtr4i 2755	. 2 ⊢ ran 2nd = V
19		df-fo 6492	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
20	5, 18, 19	mpbir2an 711	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3438  {csn 4579  ⟨cop 4585  ∪ cuni 4861  ran crn 5624   Fn wfn 6481  –onto→wfo 6484  2^nd c2nd 7930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-fun 6488  df-fn 6489  df-fo 6492  df-2nd 7932

This theorem is referenced by:  br2ndeqg  7954  2ndcof  7962  df2nd2  8039  2ndconst  8041  opco2  8064  iunfo  10452  cdaf  17975  2ndf1  18119  2ndf2  18120  2ndfcl  18122  gsum2dlem2  19868  upxp  23526  uptx  23528  cnmpt2nd  23572  uniiccdif  25495  precsexlem10  28141  precsexlem11  28142  xppreima  32602  2ndimaxp  32603  2ndresdju  32606  xppreima2  32608  2ndpreima  32664  fsuppcurry1  32681  gsummpt2d  33015  gsumpart  33023  cnre2csqima  33877  filnetlem4  36354