Theorem fo2nd 8007

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 5404	. . . . 5 ⊢ {𝑥} ∈ V
2	1	rnex 7904	. . . 4 ⊢ ran {𝑥} ∈ V
3	2	uniex 7733	. . 3 ⊢ ∪ ran {𝑥} ∈ V
4		df-2nd 7987	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5	3, 4	fnmpti 6680	. 2 ⊢ 2nd Fn V
6	4	rnmpt 5937	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
7		vex 3463	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5439	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op2nda 6217	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2744	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
11		sneq 4611	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	rneqd 5918	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
13	12	unieqd 4896	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3624	. . . . . 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 2870	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
18	6, 17	eqtr4i 2761	. 2 ⊢ ran 2nd = V
19		df-fo 6536	. 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 2108  {cab 2713  ∃wrex 3060  Vcvv 3459  {csn 4601  ⟨cop 4607  ∪ cuni 4883  ran crn 5655   Fn wfn 6525  –onto→wfo 6528  2^nd c2nd 7985

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-fun 6532  df-fn 6533  df-fo 6536  df-2nd 7987

This theorem is referenced by:  br2ndeqg  8009  2ndcof  8017  df2nd2  8096  2ndconst  8098  opco2  8121  iunfo  10551  cdaf  18061  2ndf1  18205  2ndf2  18206  2ndfcl  18208  gsum2dlem2  19950  upxp  23559  uptx  23561  cnmpt2nd  23605  uniiccdif  25529  precsexlem10  28157  precsexlem11  28158  xppreima  32569  2ndimaxp  32570  2ndresdju  32573  xppreima2  32575  2ndpreima  32631  fsuppcurry1  32648  gsummpt2d  32989  gsumpart  32997  cnre2csqima  33888  filnetlem4  36345