Theorem fo2nd 4082

Description: The 2^nd function maps the universe onto the universe.

Assertion

Ref	Expression
fo2nd	⊢ 2nd :V–onto→V

Proof of Theorem fo2nd

Step	Hyp	Ref	Expression
1		df-fo 3191	. . 3 ⊢ ({⟨x, y⟩∣y = ∪ran { x}}:V–onto→V ↔ ({⟨x, y⟩∣y = ∪ran { x}} Fn V ⋀ ran {⟨x, y⟩∣y = ∪ran { x}} = V))
2		snex 2745	. . . . . 6 ⊢ {x} ∈ V
3		rnexg 3353	. . . . . 6 ⊢ ({x} ∈ V → ran { x} ∈ V)
4	2, 3	ax-mp 7	. . . . 5 ⊢ ran { x} ∈ V
5	4	uniex 2865	. . . 4 ⊢ ∪ran { x} ∈ V
6		visset 1809	. . . . . 6 ⊢ x ∈ V
7	6	biantrur 724	. . . . 5 ⊢ (y = ∪ran { x} ↔ (x ∈ V ⋀ y = ∪ran { x}))
8	7	opabbii 2666	. . . 4 ⊢ {⟨x, y⟩∣y = ∪ran { x}} = {⟨x, y⟩∣(x ∈ V ⋀ y = ∪ran { x})}
9	5, 8	fnopab2 3610	. . 3 ⊢ {⟨x, y⟩∣y = ∪ran { x}} Fn V
10		visset 1809	. . . . . . . . 9 ⊢ y ∈ V
11	10, 10	op2nda 3444	. . . . . . . 8 ⊢ ∪ran {⟨y, y⟩} = y
12	11	eqcomi 1476	. . . . . . 7 ⊢ y = ∪ran {⟨y, y⟩}
13		opex 2777	. . . . . . . 8 ⊢ ⟨y, y⟩ ∈ V
14		sneq 2413	. . . . . . . . . . 11 ⊢ (x = ⟨y, y⟩ → {x} = {⟨y, y⟩})
15	14	rneqd 3336	. . . . . . . . . 10 ⊢ (x = ⟨y, y⟩ → ran { x} = ran {⟨y, y⟩})
16	15	unieqd 2507	. . . . . . . . 9 ⊢ (x = ⟨y, y⟩ → ∪ran { x} = ∪ran {⟨y, y⟩})
17	16	eqeq2d 1483	. . . . . . . 8 ⊢ (x = ⟨y, y⟩ → (y = ∪ran { x} ↔ y = ∪ran {⟨y, y⟩}))
18	13, 17	cla4ev 1865	. . . . . . 7 ⊢ (y = ∪ran {⟨y, y⟩} → ∃x y = ∪ran { x})
19	12, 18	ax-mp 7	. . . . . 6 ⊢ ∃x y = ∪ran { x}
20		equid 1124	. . . . . 6 ⊢ y = y
21	19, 20	2th 717	. . . . 5 ⊢ (∃x y = ∪ran { x} ↔ y = y)
22	21	abbii 1572	. . . 4 ⊢ {y∣∃x y = ∪ran { x}} = {y∣y = y}
23		rnopab 3347	. . . 4 ⊢ ran {⟨x, y⟩∣y = ∪ran { x}} = {y∣∃x y = ∪ran { x}}
24		df-v 1808	. . . 4 ⊢ V = {y∣y = y}
25	22, 23, 24	3eqtr4 1502	. . 3 ⊢ ran {⟨x, y⟩∣y = ∪ran { x}} = V
26	1, 9, 25	mpbir2an 729	. 2 ⊢ {⟨x, y⟩∣y = ∪ran { x}}:V–onto→V
27		df-2nd 4070	. . 3 ⊢ 2nd = {⟨x, y⟩∣y = ∪ran { x}}
28		foeq1 3659	. . 3 ⊢ (2nd = {⟨x, y⟩∣y = ∪ran { x}} → (2nd :V–onto→V ↔ {⟨x, y⟩∣y = ∪ran { x}}:V–onto→V))
29	27, 28	ax-mp 7	. 2 ⊢ (2nd :V–onto→V ↔ {⟨x, y⟩∣y = ∪ran { x}}:V–onto→V)
30	26, 29	mpbir 190	1 ⊢ 2nd :V–onto→V

Syntax hints:   ↔ wb 146   ⋀ wa 223   = wceq 954   ∈ wcel 956  ∃wex 978  {cab 1461  Vcvv 1807  {csn 2405  ⟨cop 2407  ∪cuni 2498  {copab 2661  ran crn 3166   Fn wfn 3172  –onto→wfo 3175  2^nd c2nd 4068

This theorem is referenced by:  2ndconst 4087  df2nd2 4117  ruclem11 7471  smfval 8176  codval 10536  idval 10537  cmpval 10538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-fo 3191  df-2nd 4070

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