Theorem fo2nd 6213

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 2763	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4215	. . . . 5 ⊢ {𝑥} ∈ V
3	2	rnex 4930	. . . 4 ⊢ ran {𝑥} ∈ V
4	3	uniex 4469	. . 3 ⊢ ∪ ran {𝑥} ∈ V
5		df-2nd 6196	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
6	4, 5	fnmpti 5383	. 2 ⊢ 2nd Fn V
7	5	rnmpt 4911	. . 3 ⊢ ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
8		vex 2763	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4259	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op2nda 5151	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2197	. . . . . 6 ⊢ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}
12		sneq 3630	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	rneqd 4892	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
14	13	unieqd 3847	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑦⟩})
15	14	eqeq2d 2205	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ ran {𝑥} ↔ 𝑦 = ∪ ran {⟨𝑦, 𝑦⟩}))
16	15	rspcev 2865	. . . . . 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 2308	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ ran {𝑥}}
20	7, 19	eqtr4i 2217	. 2 ⊢ ran 2nd = V
21		df-fo 5261	. 2 ⊢ (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
22	6, 20, 21	mpbir2an 944	1 ⊢ 2nd :V–onto→V

Syntax hints:   = wceq 1364   ∈ wcel 2164  {cab 2179  ∃wrex 2473  Vcvv 2760  {csn 3619  ⟨cop 3622  ∪ cuni 3836  ran crn 4661   Fn wfn 5250  –onto→wfo 5253  2^nd c2nd 6194

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-fun 5257  df-fn 5258  df-fo 5261  df-2nd 6196

This theorem is referenced by:  2ndcof  6219  2ndexg  6223  df2nd2  6275  2ndconst  6277  suplocexprlemmu  7780  suplocexprlemdisj  7782  suplocexprlemloc  7783  suplocexprlemub  7785  upxp  14451  uptx  14453  cnmpt2nd  14468