Theorem xpsnen 5494

Description: A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254.

Hypotheses

Ref	Expression
xpsnen.1	|- A e. _V
xpsnen.2	|- B e. _V

Assertion

Ref	Expression
xpsnen	|- (A X. {B}) ~~ A

Proof of Theorem xpsnen

Step	Hyp	Ref	Expression
1		xpsnen.1	. . 3 |- A e. _V
2		snex 3492	. . 3 |- {B} e. _V
3	1, 2	xpex 4096	. 2 |- (A X. {B}) e. _V
4		elxp 4018	. . 3 |- (y e. (A X. {B}) <-> E.xE.z(y = <.x, z>. /\ (x e. A /\ z e. {B})))
5		inteq 3217	. . . . . . . 8 |- (y = <.x, z>. -> |^|y = |^|<.x, z>.)
6	5	inteqd 3219	. . . . . . 7 |- (y = <.x, z>. -> |^||^|y = |^||^|<.x, z>.)
7		visset 2295	. . . . . . . 8 |- x e. _V
8	7	op1stb 3857	. . . . . . 7 |- |^||^|<.x, z>. = x
9	6, 8	syl6eq 1944	. . . . . 6 |- (y = <.x, z>. -> |^||^|y = x)
10	9, 7	syl6eqel 1979	. . . . 5 |- (y = <.x, z>. -> |^||^|y e. _V)
11	10	adantr 425	. . . 4 |- ((y = <.x, z>. /\ (x e. A /\ z e. {B})) -> |^||^|y e. _V)
12	11	19.23aivv 1675	. . 3 |- (E.xE.z(y = <.x, z>. /\ (x e. A /\ z e. {B})) -> |^||^|y e. _V)
13	4, 12	sylbi 216	. 2 |- (y e. (A X. {B}) -> |^||^|y e. _V)
14		opex 3527	. . 3 |- <.x, B>. e. _V
15	14	a1i 8	. 2 |- (x e. A -> <.x, B>. e. _V)
16		eleq1 1957	. . . . . 6 |- (x = |^||^|y -> (x e. _V <-> |^||^|y e. _V))
17	7, 16	mpbii 210	. . . . 5 |- (x = |^||^|y -> |^||^|y e. _V)
18		opeq1 3158	. . . . . . . . 9 |- (x = |^||^|y -> <.x, B>. = <.|^||^|y, B>.)
19	18	eqeq2d 1895	. . . . . . . 8 |- (x = |^||^|y -> (y = <.x, B>. <-> y = <.|^||^|y, B>.))
20		eleq1 1957	. . . . . . . 8 |- (x = |^||^|y -> (x e. A <-> |^||^|y e. A))
21	19, 20	anbi12d 690	. . . . . . 7 |- (x = |^||^|y -> ((y = <.x, B>. /\ x e. A) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
22	21	ceqsexgv 2393	. . . . . 6 |- (|^||^|y e. _V -> (E.x(x = |^||^|y /\ (y = <.x, B>. /\ x e. A)) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
23		ancom 482	. . . . . . . . . . 11 |- (((y = <.x, z>. /\ x e. A) /\ z e. {B}) <-> (z e. {B} /\ (y = <.x, z>. /\ x e. A)))
24		anass 487	. . . . . . . . . . 11 |- (((y = <.x, z>. /\ x e. A) /\ z e. {B}) <-> (y = <.x, z>. /\ (x e. A /\ z e. {B})))
25		elsn 3058	. . . . . . . . . . . 12 |- (z e. {B} <-> z = B)
26	25	anbi1i 539	. . . . . . . . . . 11 |- ((z e. {B} /\ (y = <.x, z>. /\ x e. A)) <-> (z = B /\ (y = <.x, z>. /\ x e. A)))
27	23, 24, 26	3bitr3i 198	. . . . . . . . . 10 |- ((y = <.x, z>. /\ (x e. A /\ z e. {B})) <-> (z = B /\ (y = <.x, z>. /\ x e. A)))
28	27	exbii 1398	. . . . . . . . 9 |- (E.z(y = <.x, z>. /\ (x e. A /\ z e. {B})) <-> E.z(z = B /\ (y = <.x, z>. /\ x e. A)))
29		xpsnen.2	. . . . . . . . . 10 |- B e. _V
30		opeq2 3159	. . . . . . . . . . . 12 |- (z = B -> <.x, z>. = <.x, B>.)
31	30	eqeq2d 1895	. . . . . . . . . . 11 |- (z = B -> (y = <.x, z>. <-> y = <.x, B>.))
32	31	anbi1d 679	. . . . . . . . . 10 |- (z = B -> ((y = <.x, z>. /\ x e. A) <-> (y = <.x, B>. /\ x e. A)))
33	29, 32	ceqsexv 2325	. . . . . . . . 9 |- (E.z(z = B /\ (y = <.x, z>. /\ x e. A)) <-> (y = <.x, B>. /\ x e. A))
34		inteq 3217	. . . . . . . . . . . . . 14 |- (y = <.x, B>. -> |^|y = |^|<.x, B>.)
35	34	inteqd 3219	. . . . . . . . . . . . 13 |- (y = <.x, B>. -> |^||^|y = |^||^|<.x, B>.)
36	7	op1stb 3857	. . . . . . . . . . . . 13 |- |^||^|<.x, B>. = x
37	35, 36	syl6req 1945	. . . . . . . . . . . 12 |- (y = <.x, B>. -> x = |^||^|y)
38	37	pm4.71ri 700	. . . . . . . . . . 11 |- (y = <.x, B>. <-> (x = |^||^|y /\ y = <.x, B>.))
39	38	anbi1i 539	. . . . . . . . . 10 |- ((y = <.x, B>. /\ x e. A) <-> ((x = |^||^|y /\ y = <.x, B>.) /\ x e. A))
40		anass 487	. . . . . . . . . 10 |- (((x = |^||^|y /\ y = <.x, B>.) /\ x e. A) <-> (x = |^||^|y /\ (y = <.x, B>. /\ x e. A)))
41	39, 40	bitri 190	. . . . . . . . 9 |- ((y = <.x, B>. /\ x e. A) <-> (x = |^||^|y /\ (y = <.x, B>. /\ x e. A)))
42	28, 33, 41	3bitri 194	. . . . . . . 8 |- (E.z(y = <.x, z>. /\ (x e. A /\ z e. {B})) <-> (x = |^||^|y /\ (y = <.x, B>. /\ x e. A)))
43	42	exbii 1398	. . . . . . 7 |- (E.xE.z(y = <.x, z>. /\ (x e. A /\ z e. {B})) <-> E.x(x = |^||^|y /\ (y = <.x, B>. /\ x e. A)))
44	4, 43	bitri 190	. . . . . 6 |- (y e. (A X. {B}) <-> E.x(x = |^||^|y /\ (y = <.x, B>. /\ x e. A)))
45	22, 44	syl5bb 591	. . . . 5 |- (|^||^|y e. _V -> (y e. (A X. {B}) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
46	17, 45	syl 12	. . . 4 |- (x = |^||^|y -> (y e. (A X. {B}) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
47	46	pm5.32ri 708	. . 3 |- ((y e. (A X. {B}) /\ x = |^||^|y) <-> ((y = <.|^||^|y, B>. /\ |^||^|y e. A) /\ x = |^||^|y))
48	37	adantr 425	. . . . 5 |- ((y = <.x, B>. /\ x e. A) -> x = |^||^|y)
49	48	pm4.71i 699	. . . 4 |- ((y = <.x, B>. /\ x e. A) <-> ((y = <.x, B>. /\ x e. A) /\ x = |^||^|y))
50	21	pm5.32ri 708	. . . 4 |- (((y = <.x, B>. /\ x e. A) /\ x = |^||^|y) <-> ((y = <.|^||^|y, B>. /\ |^||^|y e. A) /\ x = |^||^|y))
51	49, 50	bitr2i 191	. . 3 |- (((y = <.|^||^|y, B>. /\ |^||^|y e. A) /\ x = |^||^|y) <-> (y = <.x, B>. /\ x e. A))
52		ancom 482	. . 3 |- ((y = <.x, B>. /\ x e. A) <-> (x e. A /\ y = <.x, B>.))
53	47, 51, 52	3bitri 194	. 2 |- ((y e. (A X. {B}) /\ x = |^||^|y) <-> (x e. A /\ y = <.x, B>.))
54	3, 13, 15, 53	en2 5461	1 |- (A X. {B}) ~~ A

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292  {csn 3044  <.cop 3046  |^|cint 3214   class class class wbr 3338   X. cxp 3984   ~~ cen 5423

This theorem is referenced by:  xpsneng 5495  endisj 5496  xpdom3 5504  unxpdom2 5997  sucxpdom 5998  uncdadom 6069  cdaun 6070  pm110.643 6072  cdaen 6073  cda0en 6075  cda1en 6076  xp1en 6077  cdacomen 6079  cdaassen 6080  mapcdaen 6082  cdadom1 6083  xpnnen 8768

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-en 5427

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