Theorem fo2nd 6314

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 2803	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4271	. . . . 5 ⊢ {𝑥} ∈ V
3	2	rnex 4996	. . . 4 ⊢ ran {𝑥} ∈ V
4	3	uniex 4530	. . 3 ⊢ ∪ ran {𝑥} ∈ V
5		df-2nd 6297	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
6	4, 5	fnmpti 5456	. 2 ⊢ 2nd Fn V
7	5	rnmpt 4976	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
8		vex 2803	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4317	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op2nda 5217	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2233	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
12		sneq 3678	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	rneqd 4957	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
14	13	unieqd 3900	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
15	14	eqeq2d 2241	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}))
16	15	rspcev 2908	. . . . . 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 2344	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
20	7, 19	eqtr4i 2253	. 2 ⊢ ran 2nd = V
21		df-fo 5328	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
22	6, 20, 21	mpbir2an 948	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1395   ∈ wcel 2200  {cab 2215  ∃wrex 2509  Vcvv 2800  {csn 3667  ⟨cop 3670  ∪ cuni 3889  ran crn 4722   Fn wfn 5317  –onto→wfo 5320  2^nd c2nd 6295

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-fun 5324  df-fn 5325  df-fo 5328  df-2nd 6297

This theorem is referenced by:  2ndcof  6320  2ndexg  6324  df2nd2  6378  2ndconst  6380  suplocexprlemmu  7926  suplocexprlemdisj  7928  suplocexprlemloc  7929  suplocexprlemub  7931  upxp  14983  uptx  14985  cnmpt2nd  15000