Theorem fo2nd 5727

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 2554	. . . . . 6 ⊢ x ∈ V
2		snexgOLD 3926	. . . . . 6 ⊢ (x ∈ V → {x} ∈ V)
3	1, 2	ax-mp 7	. . . . 5 ⊢ {x} ∈ V
4	3	rnex 4542	. . . 4 ⊢ ran {x} ∈ V
5	4	uniex 4140	. . 3 ⊢ ∪ ran {x} ∈ V
6		df-2nd 5710	. . 3 ⊢ 2nd = (x ∈ V ↦ ∪ ran {x})
7	5, 6	fnmpti 4970	. 2 ⊢ 2nd Fn V
8	6	rnmpt 4525	. . 3 ⊢ ran 2nd = {y ∣ ∃x ∈ V y = ∪ ran {x}}
9		vex 2554	. . . . 5 ⊢ y ∈ V
10	9, 9	opex 3957	. . . . . 6 ⊢ ⟨y, y⟩ ∈ V
11	9, 9	op2nda 4748	. . . . . . 7 ⊢ ∪ ran {⟨y, y⟩} = y
12	11	eqcomi 2041	. . . . . 6 ⊢ y = ∪ ran {⟨y, y⟩}
13		sneq 3378	. . . . . . . . . 10 ⊢ (x = ⟨y, y⟩ → {x} = {⟨y, y⟩})
14	13	rneqd 4506	. . . . . . . . 9 ⊢ (x = ⟨y, y⟩ → ran {x} = ran {⟨y, y⟩})
15	14	unieqd 3582	. . . . . . . 8 ⊢ (x = ⟨y, y⟩ → ∪ ran {x} = ∪ ran {⟨y, y⟩})
16	15	eqeq2d 2048	. . . . . . 7 ⊢ (x = ⟨y, y⟩ → (y = ∪ ran {x} ↔ y = ∪ ran {⟨y, y⟩}))
17	16	rspcev 2650	. . . . . 6 ⊢ ((⟨y, y⟩ ∈ V ∧ y = ∪ ran {⟨y, y⟩}) → ∃x ∈ V y = ∪ ran {x})
18	10, 12, 17	mp2an 402	. . . . 5 ⊢ ∃x ∈ V y = ∪ ran {x}
19	9, 18	2th 163	. . . 4 ⊢ (y ∈ V ↔ ∃x ∈ V y = ∪ ran {x})
20	19	abbi2i 2149	. . 3 ⊢ V = {y ∣ ∃x ∈ V y = ∪ ran {x}}
21	8, 20	eqtr4i 2060	. 2 ⊢ ran 2nd = V
22		df-fo 4851	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
23	7, 21, 22	mpbir2an 848	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551  {csn 3367  ⟨cop 3370  ∪ cuni 3571  ran crn 4289   Fn wfn 4840  –onto→wfo 4843  2^nd c2nd 5708

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-fo 4851  df-2nd 5710

This theorem is referenced by:  2ndcof  5733  2ndexg  5737  df2nd2  5783  2ndconst  5785