Theorem csucex 6260

Description: The function mapping x to its cardinal successor exists. (Contributed by Scott Fenton, 30-Jul-2019.)

Assertion

Ref	Expression
csucex	⊢ (x ∈ V ↦ (x +c 1c)) ∈ V

Proof of Theorem csucex

Dummy variables y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brcnv 4893	. . . . . . . . . 10 ⊢ (y◡1st w ↔ w1st y)
2		vex 2863	. . . . . . . . . . 11 ⊢ y ∈ V
3	2	br1st 4859	. . . . . . . . . 10 ⊢ (w1st y ↔ ∃x w = ⟨y, x⟩)
4	1, 3	bitri 240	. . . . . . . . 9 ⊢ (y◡1st w ↔ ∃x w = ⟨y, x⟩)
5	4	anbi1i 676	. . . . . . . 8 ⊢ ((y◡1st w ∧ w( AddC ↾ (V × {1c}))z) ↔ (∃x w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z))
6		19.41v 1901	. . . . . . . 8 ⊢ (∃x(w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z) ↔ (∃x w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z))
7	5, 6	bitr4i 243	. . . . . . 7 ⊢ ((y◡1st w ∧ w( AddC ↾ (V × {1c}))z) ↔ ∃x(w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z))
8	7	exbii 1582	. . . . . 6 ⊢ (∃w(y◡1st w ∧ w( AddC ↾ (V × {1c}))z) ↔ ∃w∃x(w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z))
9		excom 1741	. . . . . . 7 ⊢ (∃w∃x(w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z) ↔ ∃x∃w(w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z))
10		vex 2863	. . . . . . . . . 10 ⊢ x ∈ V
11	2, 10	opex 4589	. . . . . . . . 9 ⊢ ⟨y, x⟩ ∈ V
12		breq1 4643	. . . . . . . . . 10 ⊢ (w = ⟨y, x⟩ → (w( AddC ↾ (V × {1c}))z ↔ ⟨y, x⟩( AddC ↾ (V × {1c}))z))
13		brres 4950	. . . . . . . . . . 11 ⊢ (⟨y, x⟩( AddC ↾ (V × {1c}))z ↔ (⟨y, x⟩ AddC z ∧ ⟨y, x⟩ ∈ (V × {1c})))
14	2, 10	braddcfn 5827	. . . . . . . . . . . 12 ⊢ (⟨y, x⟩ AddC z ↔ (y +c x) = z)
15		opelxp 4812	. . . . . . . . . . . . . 14 ⊢ (⟨y, x⟩ ∈ (V × {1c}) ↔ (y ∈ V ∧ x ∈ {1c}))
16	2, 15	mpbiran 884	. . . . . . . . . . . . 13 ⊢ (⟨y, x⟩ ∈ (V × {1c}) ↔ x ∈ {1c})
17		elsn 3749	. . . . . . . . . . . . 13 ⊢ (x ∈ {1c} ↔ x = 1c)
18	16, 17	bitri 240	. . . . . . . . . . . 12 ⊢ (⟨y, x⟩ ∈ (V × {1c}) ↔ x = 1c)
19	14, 18	anbi12ci 679	. . . . . . . . . . 11 ⊢ ((⟨y, x⟩ AddC z ∧ ⟨y, x⟩ ∈ (V × {1c})) ↔ (x = 1c ∧ (y +c x) = z))
20	13, 19	bitri 240	. . . . . . . . . 10 ⊢ (⟨y, x⟩( AddC ↾ (V × {1c}))z ↔ (x = 1c ∧ (y +c x) = z))
21	12, 20	syl6bb 252	. . . . . . . . 9 ⊢ (w = ⟨y, x⟩ → (w( AddC ↾ (V × {1c}))z ↔ (x = 1c ∧ (y +c x) = z)))
22	11, 21	ceqsexv 2895	. . . . . . . 8 ⊢ (∃w(w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z) ↔ (x = 1c ∧ (y +c x) = z))
23	22	exbii 1582	. . . . . . 7 ⊢ (∃x∃w(w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z) ↔ ∃x(x = 1c ∧ (y +c x) = z))
24	9, 23	bitri 240	. . . . . 6 ⊢ (∃w∃x(w = ⟨y, x⟩ ∧ w( AddC ↾ (V × {1c}))z) ↔ ∃x(x = 1c ∧ (y +c x) = z))
25	8, 24	bitri 240	. . . . 5 ⊢ (∃w(y◡1st w ∧ w( AddC ↾ (V × {1c}))z) ↔ ∃x(x = 1c ∧ (y +c x) = z))
26		1cex 4143	. . . . . 6 ⊢ 1c ∈ V
27		addceq2 4385	. . . . . . 7 ⊢ (x = 1c → (y +c x) = (y +c 1c))
28	27	eqeq1d 2361	. . . . . 6 ⊢ (x = 1c → ((y +c x) = z ↔ (y +c 1c) = z))
29	26, 28	ceqsexv 2895	. . . . 5 ⊢ (∃x(x = 1c ∧ (y +c x) = z) ↔ (y +c 1c) = z)
30	25, 29	bitri 240	. . . 4 ⊢ (∃w(y◡1st w ∧ w( AddC ↾ (V × {1c}))z) ↔ (y +c 1c) = z)
31		opelco 4885	. . . 4 ⊢ (⟨y, z⟩ ∈ (( AddC ↾ (V × {1c})) ∘ ◡1st ) ↔ ∃w(y◡1st w ∧ w( AddC ↾ (V × {1c}))z))
32		mptv 5719	. . . . . 6 ⊢ (x ∈ V ↦ (x +c 1c)) = {⟨x, w⟩ ∣ w = (x +c 1c)}
33	32	eleq2i 2417	. . . . 5 ⊢ (⟨y, z⟩ ∈ (x ∈ V ↦ (x +c 1c)) ↔ ⟨y, z⟩ ∈ {⟨x, w⟩ ∣ w = (x +c 1c)})
34		vex 2863	. . . . . 6 ⊢ z ∈ V
35		addceq1 4384	. . . . . . 7 ⊢ (x = y → (x +c 1c) = (y +c 1c))
36	35	eqeq2d 2364	. . . . . 6 ⊢ (x = y → (w = (x +c 1c) ↔ w = (y +c 1c)))
37		eqeq1 2359	. . . . . . 7 ⊢ (w = z → (w = (y +c 1c) ↔ z = (y +c 1c)))
38		eqcom 2355	. . . . . . 7 ⊢ (z = (y +c 1c) ↔ (y +c 1c) = z)
39	37, 38	syl6bb 252	. . . . . 6 ⊢ (w = z → (w = (y +c 1c) ↔ (y +c 1c) = z))
40	2, 34, 36, 39	opelopab 4709	. . . . 5 ⊢ (⟨y, z⟩ ∈ {⟨x, w⟩ ∣ w = (x +c 1c)} ↔ (y +c 1c) = z)
41	33, 40	bitri 240	. . . 4 ⊢ (⟨y, z⟩ ∈ (x ∈ V ↦ (x +c 1c)) ↔ (y +c 1c) = z)
42	30, 31, 41	3bitr4ri 269	. . 3 ⊢ (⟨y, z⟩ ∈ (x ∈ V ↦ (x +c 1c)) ↔ ⟨y, z⟩ ∈ (( AddC ↾ (V × {1c})) ∘ ◡1st ))
43	42	eqrelriv 4851	. 2 ⊢ (x ∈ V ↦ (x +c 1c)) = (( AddC ↾ (V × {1c})) ∘ ◡1st )
44		addcfnex 5825	. . . 4 ⊢ AddC ∈ V
45		vvex 4110	. . . . 5 ⊢ V ∈ V
46		snex 4112	. . . . 5 ⊢ {1c} ∈ V
47	45, 46	xpex 5116	. . . 4 ⊢ (V × {1c}) ∈ V
48	44, 47	resex 5118	. . 3 ⊢ ( AddC ↾ (V × {1c})) ∈ V
49		1stex 4740	. . . 4 ⊢ 1st ∈ V
50	49	cnvex 5103	. . 3 ⊢ ◡1st ∈ V
51	48, 50	coex 4751	. 2 ⊢ (( AddC ↾ (V × {1c})) ∘ ◡1st ) ∈ V
52	43, 51	eqeltri 2423	1 ⊢ (x ∈ V ↦ (x +c 1c)) ∈ V

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {csn 3738  1_cc1c 4135   +_c cplc 4376  ⟨cop 4562  {copab 4623   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722   × cxp 4771  ^◡ccnv 4772   ↾ cres 4775   ↦ cmpt 5652   AddC caddcfn 5746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-cup 5743  df-disj 5745  df-addcfn 5747  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759

This theorem is referenced by:  nnltp1clem1  6262  frecexg  6313  frecxp  6315  dmfrec  6317  fnfreclem2  6319  fnfreclem3  6320  frec0  6322  frecsuc  6323