Theorem fo2nd 7992

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 5392	. . . . 5 ⊢ {𝑥} ∈ V
2	1	rnex 7889	. . . 4 ⊢ ran {𝑥} ∈ V
3	2	uniex 7720	. . 3 ⊢ ∪ ran {𝑥} ∈ V
4		df-2nd 7972	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5	3, 4	fnmpti 6664	. 2 ⊢ 2nd Fn V
6	4	rnmpt 5924	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
7		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5427	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op2nda 6204	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2739	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
11		sneq 4602	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	rneqd 5905	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
13	12	unieqd 4887	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3614	. . . . . 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 2864	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
18	6, 17	eqtr4i 2756	. 2 ⊢ ran 2nd = V
19		df-fo 6520	. 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 2708  ∃wrex 3054  Vcvv 3450  {csn 4592  ⟨cop 4598  ∪ cuni 4874  ran crn 5642   Fn wfn 6509  –onto→wfo 6512  2^nd c2nd 7970

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-fun 6516  df-fn 6517  df-fo 6520  df-2nd 7972

This theorem is referenced by:  br2ndeqg  7994  2ndcof  8002  df2nd2  8081  2ndconst  8083  opco2  8106  iunfo  10499  cdaf  18019  2ndf1  18163  2ndf2  18164  2ndfcl  18166  gsum2dlem2  19908  upxp  23517  uptx  23519  cnmpt2nd  23563  uniiccdif  25486  precsexlem10  28125  precsexlem11  28126  xppreima  32576  2ndimaxp  32577  2ndresdju  32580  xppreima2  32582  2ndpreima  32638  fsuppcurry1  32655  gsummpt2d  32996  gsumpart  33004  cnre2csqima  33908  filnetlem4  36376