Theorem fo2nd 5785

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 2560	. . . . . 6 ⊢ 𝑥 ∈ V
2		snexgOLD 3935	. . . . . 6 ⊢ (𝑥 ∈ V → {𝑥} ∈ V)
3	1, 2	ax-mp 7	. . . . 5 ⊢ {𝑥} ∈ V
4	3	rnex 4599	. . . 4 ⊢ ran {𝑥} ∈ V
5	4	uniex 4174	. . 3 ⊢ ∪ ran {𝑥} ∈ V
6		df-2nd 5768	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
7	5, 6	fnmpti 5027	. 2 ⊢ 2nd Fn V
8	6	rnmpt 4582	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
9		vex 2560	. . . . 5 ⊢ 𝑦 ∈ V
10	9, 9	opex 3966	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
11	9, 9	op2nda 4805	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
12	11	eqcomi 2044	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
13		sneq 3386	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
14	13	rneqd 4563	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
15	14	unieqd 3591	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
16	15	eqeq2d 2051	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}))
17	16	rspcev 2656	. . . . . 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 2152	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
21	8, 20	eqtr4i 2063	. 2 ⊢ ran 2nd = V
22		df-fo 4908	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
23	7, 21, 22	mpbir2an 849	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1243   ∈ wcel 1393  {cab 2026  ∃wrex 2307  Vcvv 2557  {csn 3375  ⟨cop 3378  ∪ cuni 3580  ran crn 4346   Fn wfn 4897  –onto→wfo 4900  2^nd c2nd 5766

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-fo 4908  df-2nd 5768

This theorem is referenced by:  2ndcof  5791  2ndexg  5795  df2nd2  5841  2ndconst  5843