Theorem fo2nd 5943

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 2623	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4026	. . . . 5 ⊢ {𝑥} ∈ V
3	2	rnex 4713	. . . 4 ⊢ ran {𝑥} ∈ V
4	3	uniex 4273	. . 3 ⊢ ∪ ran {𝑥} ∈ V
5		df-2nd 5926	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
6	4, 5	fnmpti 5155	. 2 ⊢ 2nd Fn V
7	5	rnmpt 4696	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
8		vex 2623	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4065	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op2nda 4928	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2093	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
12		sneq 3461	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	rneqd 4677	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
14	13	unieqd 3670	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
15	14	eqeq2d 2100	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}))
16	15	rspcev 2723	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
17	9, 11, 16	mp2an 418	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}
18	8, 17	2th 173	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
19	18	abbi2i 2203	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
20	7, 19	eqtr4i 2112	. 2 ⊢ ran 2nd = V
21		df-fo 5034	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
22	6, 20, 21	mpbir2an 889	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1290   ∈ wcel 1439  {cab 2075  ∃wrex 2361  Vcvv 2620  {csn 3450  ⟨cop 3453  ∪ cuni 3659  ran crn 4453   Fn wfn 5023  –onto→wfo 5026  2^nd c2nd 5924

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-fun 5030  df-fn 5031  df-fo 5034  df-2nd 5926

This theorem is referenced by:  2ndcof  5949  2ndexg  5953  df2nd2  5999  2ndconst  6001