Theorem fo2nd 7189

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		snex 4908	. . . . 5 ⊢ {𝑥} ∈ V
2	1	rnex 7100	. . . 4 ⊢ ran {𝑥} ∈ V
3	2	uniex 6953	. . 3 ⊢ ∪ ran {𝑥} ∈ V
4		df-2nd 7169	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5	3, 4	fnmpti 6022	. 2 ⊢ 2nd Fn V
6	4	rnmpt 5371	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
7		vex 3203	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 4932	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op2nda 5620	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2631	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
11		sneq 4187	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	rneqd 5353	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
13	12	unieqd 4446	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
14	13	eqeq2d 2632	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}))
15	14	rspcev 3309	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
16	8, 10, 15	mp2an 708	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}
17	7, 16	2th 254	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
18	17	abbi2i 2738	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
19	6, 18	eqtr4i 2647	. 2 ⊢ ran 2nd = V
20		df-fo 5894	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
21	5, 19, 20	mpbir2an 955	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  Vcvv 3200  {csn 4177  ⟨cop 4183  ∪ cuni 4436  ran crn 5115   Fn wfn 5883  –onto→wfo 5886  2^nd c2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-fo 5894  df-2nd 7169

This theorem is referenced by:  2ndcof  7197  df2nd2  7264  2ndconst  7266  iunfo  9361  cdaf  16700  2ndf1  16835  2ndf2  16836  2ndfcl  16838  gsum2dlem2  18370  upxp  21426  uptx  21428  cnmpt2nd  21472  uniiccdif  23346  xppreima  29449  xppreima2  29450  2ndpreima  29485  gsummpt2d  29781  cnre2csqima  29957  br2ndeq  31671  br2ndeqg  31673  filnetlem4  32376