Theorem fo2nd 6126

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 2729	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4164	. . . . 5 ⊢ {𝑥} ∈ V
3	2	rnex 4871	. . . 4 ⊢ ran {𝑥} ∈ V
4	3	uniex 4415	. . 3 ⊢ ∪ ran {𝑥} ∈ V
5		df-2nd 6109	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
6	4, 5	fnmpti 5316	. 2 ⊢ 2nd Fn V
7	5	rnmpt 4852	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
8		vex 2729	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4207	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op2nda 5088	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2169	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
12		sneq 3587	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	rneqd 4833	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
14	13	unieqd 3800	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
15	14	eqeq2d 2177	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}))
16	15	rspcev 2830	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
17	9, 11, 16	mp2an 423	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}
18	8, 17	2th 173	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥})
19	18	abbi2i 2281	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
20	7, 19	eqtr4i 2189	. 2 ⊢ ran 2nd = V
21		df-fo 5194	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
22	6, 20, 21	mpbir2an 932	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1343   ∈ wcel 2136  {cab 2151  ∃wrex 2445  Vcvv 2726  {csn 3576  ⟨cop 3579  ∪ cuni 3789  ran crn 4605   Fn wfn 5183  –onto→wfo 5186  2^nd c2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-fo 5194  df-2nd 6109

This theorem is referenced by:  2ndcof  6132  2ndexg  6136  df2nd2  6188  2ndconst  6190  suplocexprlemmu  7659  suplocexprlemdisj  7661  suplocexprlemloc  7662  suplocexprlemub  7664  upxp  12922  uptx  12924  cnmpt2nd  12939