Theorem fo2nd 6330

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		vex 2806	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4281	. . . . 5 ⊢ {𝑥} ∈ V
3	2	rnex 5006	. . . 4 ⊢ ran {𝑥} ∈ V
4	3	uniex 4540	. . 3 ⊢ ∪ ran {𝑥} ∈ V
5		df-2nd 6313	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
6	4, 5	fnmpti 5468	. 2 ⊢ 2nd Fn V
7	5	rnmpt 4986	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
8		vex 2806	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4327	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op2nda 5228	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2235	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
12		sneq 3684	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	rneqd 4967	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
14	13	unieqd 3909	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
15	14	eqeq2d 2243	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}))
16	15	rspcev 2911	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
17	9, 11, 16	mp2an 426	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}
18	8, 17	2th 174	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
19	18	abbi2i 2346	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
20	7, 19	eqtr4i 2255	. 2 ⊢ ran 2nd = V
21		df-fo 5339	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
22	6, 20, 21	mpbir2an 951	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1398   ∈ wcel 2202  {cab 2217  ∃wrex 2512  Vcvv 2803  {csn 3673  ⟨cop 3676  ∪ cuni 3898  ran crn 4732   Fn wfn 5328  –onto→wfo 5331  2^nd c2nd 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-fo 5339  df-2nd 6313

This theorem is referenced by:  2ndcof  6336  2ndexg  6340  df2nd2  6394  2ndconst  6396  suplocexprlemmu  7981  suplocexprlemdisj  7983  suplocexprlemloc  7984  suplocexprlemub  7986  upxp  15063  uptx  15065  cnmpt2nd  15080