Theorem csucex 6259

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

Assertion

Ref	Expression
csucex	|- (x e. _V |-> (x +o 1c)) e. _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 4892	. . . . . . . . . 10 |- (y`'1stw <-> w1sty)
2		vex 2862	. . . . . . . . . . 11 |- y e. _V
3	2	br1st 4858	. . . . . . . . . 10 |- (w1sty <-> E.x w = <.y, x>.)
4	1, 3	bitri 240	. . . . . . . . 9 |- (y`'1stw <-> E.x w = <.y, x>.)
5	4	anbi1i 676	. . . . . . . 8 |- ((y`'1stw /\ w( AddC |` (_V X. {1c}))z) <-> (E.x w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z))
6		19.41v 1901	. . . . . . . 8 |- (E.x(w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z) <-> (E.x w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z))
7	5, 6	bitr4i 243	. . . . . . 7 |- ((y`'1stw /\ w( AddC |` (_V X. {1c}))z) <-> E.x(w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z))
8	7	exbii 1582	. . . . . 6 |- (E.w(y`'1stw /\ w( AddC |` (_V X. {1c}))z) <-> E.wE.x(w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z))
9		excom 1741	. . . . . . 7 |- (E.wE.x(w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z) <-> E.xE.w(w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z))
10		vex 2862	. . . . . . . . . 10 |- x e. _V
11	2, 10	opex 4588	. . . . . . . . 9 |- <.y, x>. e. _V
12		breq1 4642	. . . . . . . . . 10 |- (w = <.y, x>. -> (w( AddC |` (_V X. {1c}))z <-> <.y, x>.( AddC |` (_V X. {1c}))z))
13		brres 4949	. . . . . . . . . . 11 |- (<.y, x>.( AddC |` (_V X. {1c}))z <-> (<.y, x>. AddC z /\ <.y, x>. e. (_V X. {1c})))
14	2, 10	braddcfn 5826	. . . . . . . . . . . 12 |- (<.y, x>. AddC z <-> (y +o x) = z)
15		opelxp 4811	. . . . . . . . . . . . . 14 |- (<.y, x>. e. (_V X. {1c}) <-> (y e. _V /\ x e. {1c}))
16	2, 15	mpbiran 884	. . . . . . . . . . . . 13 |- (<.y, x>. e. (_V X. {1c}) <-> x e. {1c})
17		elsn 3748	. . . . . . . . . . . . 13 |- (x e. {1c} <-> x = 1c)
18	16, 17	bitri 240	. . . . . . . . . . . 12 |- (<.y, x>. e. (_V X. {1c}) <-> x = 1c)
19	14, 18	anbi12ci 679	. . . . . . . . . . 11 |- ((<.y, x>. AddC z /\ <.y, x>. e. (_V X. {1c})) <-> (x = 1c /\ (y +o x) = z))
20	13, 19	bitri 240	. . . . . . . . . 10 |- (<.y, x>.( AddC |` (_V X. {1c}))z <-> (x = 1c /\ (y +o x) = z))
21	12, 20	syl6bb 252	. . . . . . . . 9 |- (w = <.y, x>. -> (w( AddC |` (_V X. {1c}))z <-> (x = 1c /\ (y +o x) = z)))
22	11, 21	ceqsexv 2894	. . . . . . . 8 |- (E.w(w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z) <-> (x = 1c /\ (y +o x) = z))
23	22	exbii 1582	. . . . . . 7 |- (E.xE.w(w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z) <-> E.x(x = 1c /\ (y +o x) = z))
24	9, 23	bitri 240	. . . . . 6 |- (E.wE.x(w = <.y, x>. /\ w( AddC |` (_V X. {1c}))z) <-> E.x(x = 1c /\ (y +o x) = z))
25	8, 24	bitri 240	. . . . 5 |- (E.w(y`'1stw /\ w( AddC |` (_V X. {1c}))z) <-> E.x(x = 1c /\ (y +o x) = z))
26		1cex 4142	. . . . . 6 |- 1c e. _V
27		addceq2 4384	. . . . . . 7 |- (x = 1c -> (y +o x) = (y +o 1c))
28	27	eqeq1d 2361	. . . . . 6 |- (x = 1c -> ((y +o x) = z <-> (y +o 1c) = z))
29	26, 28	ceqsexv 2894	. . . . 5 |- (E.x(x = 1c /\ (y +o x) = z) <-> (y +o 1c) = z)
30	25, 29	bitri 240	. . . 4 |- (E.w(y`'1stw /\ w( AddC |` (_V X. {1c}))z) <-> (y +o 1c) = z)
31		opelco 4884	. . . 4 |- (<.y, z>. e. (( AddC |` (_V X. {1c})) o. `'1st) <-> E.w(y`'1stw /\ w( AddC |` (_V X. {1c}))z))
32		mptv 5718	. . . . . 6 |- (x e. _V |-> (x +o 1c)) = {<.x, w>. | w = (x +o 1c)}
33	32	eleq2i 2417	. . . . 5 |- (<.y, z>. e. (x e. _V |-> (x +o 1c)) <-> <.y, z>. e. {<.x, w>. | w = (x +o 1c)})
34		vex 2862	. . . . . 6 |- z e. _V
35		addceq1 4383	. . . . . . 7 |- (x = y -> (x +o 1c) = (y +o 1c))
36	35	eqeq2d 2364	. . . . . 6 |- (x = y -> (w = (x +o 1c) <-> w = (y +o 1c)))
37		eqeq1 2359	. . . . . . 7 |- (w = z -> (w = (y +o 1c) <-> z = (y +o 1c)))
38		eqcom 2355	. . . . . . 7 |- (z = (y +o 1c) <-> (y +o 1c) = z)
39	37, 38	syl6bb 252	. . . . . 6 |- (w = z -> (w = (y +o 1c) <-> (y +o 1c) = z))
40	2, 34, 36, 39	opelopab 4708	. . . . 5 |- (<.y, z>. e. {<.x, w>. | w = (x +o 1c)} <-> (y +o 1c) = z)
41	33, 40	bitri 240	. . . 4 |- (<.y, z>. e. (x e. _V |-> (x +o 1c)) <-> (y +o 1c) = z)
42	30, 31, 41	3bitr4ri 269	. . 3 |- (<.y, z>. e. (x e. _V |-> (x +o 1c)) <-> <.y, z>. e. (( AddC |` (_V X. {1c})) o. `'1st))
43	42	eqrelriv 4850	. 2 |- (x e. _V |-> (x +o 1c)) = (( AddC |` (_V X. {1c})) o. `'1st)
44		addcfnex 5824	. . . 4 |- AddC e. _V
45		vvex 4109	. . . . 5 |- _V e. _V
46		snex 4111	. . . . 5 |- {1c} e. _V
47	45, 46	xpex 5115	. . . 4 |- (_V X. {1c}) e. _V
48	44, 47	resex 5117	. . 3 |- ( AddC |` (_V X. {1c})) e. _V
49		1stex 4739	. . . 4 |- 1st e. _V
50	49	cnvex 5102	. . 3 |- `'1st e. _V
51	48, 50	coex 4750	. 2 |- (( AddC |` (_V X. {1c})) o. `'1st) e. _V
52	43, 51	eqeltri 2423	1 |- (x e. _V |-> (x +o 1c)) e. _V

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859  {csn 3737  1_cc1c 4134   +o cplc 4375  <.cop 4561  {copab 4622   class class class wbr 4639  1stc1st 4717   o. ccom 4721   X. cxp 4770  `'ccnv 4771   |` cres 4774   |-> cmpt 5651   AddC caddcfn 5745

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758

This theorem is referenced by:  nnltp1clem1  6261  frecexg  6312  frecxp  6314  dmfrec  6316  fnfreclem2  6318  fnfreclem3  6319  frec0  6321  frecsuc  6322