Theorem xpsnen 4435

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 2750	. . 3 |- {B} e. V
3	1, 2	xpex 3260	. 2 |- (A X. {B}) e. V
4		elxp 3202	. . 3 |- (y e. (A X. {B}) <-> E.xE.z(y = <.x, z>. /\ (x e. A /\ z e. {B})))
5		inteq 2536	. . . . . . . 8 |- (y = <.x, z>. -> |^|y = |^|<.x, z>.)
6	5	inteqd 2538	. . . . . . 7 |- (y = <.x, z>. -> |^||^|y = |^||^|<.x, z>.)
7		visset 1813	. . . . . . . 8 |- x e. V
8	7	op1stb 2913	. . . . . . 7 |- |^||^|<.x, z>. = x
9	6, 8	syl6eq 1523	. . . . . 6 |- (y = <.x, z>. -> |^||^|y = x)
10	9, 7	syl6eqel 1556	. . . . 5 |- (y = <.x, z>. -> |^||^|y e. V)
11	10	adantr 389	. . . 4 |- ((y = <.x, z>. /\ (x e. A /\ z e. {B})) -> |^||^|y e. V)
12	11	19.23aivv 1296	. . 3 |- (E.xE.z(y = <.x, z>. /\ (x e. A /\ z e. {B})) -> |^||^|y e. V)
13	4, 12	sylbi 199	. 2 |- (y e. (A X. {B}) -> |^||^|y e. V)
14		opex 2782	. . 3 |- <.x, B>. e. V
15	14	a1i 8	. 2 |- (x e. A -> <.x, B>. e. V)
16		eleq1 1534	. . . . . 6 |- (x = |^||^|y -> (x e. V <-> |^||^|y e. V))
17	7, 16	mpbii 193	. . . . 5 |- (x = |^||^|y -> |^||^|y e. V)
18		opeq1 2487	. . . . . . . . 9 |- (x = |^||^|y -> <.x, B>. = <.|^||^|y, B>.)
19	18	eqeq2d 1486	. . . . . . . 8 |- (x = |^||^|y -> (y = <.x, B>. <-> y = <.|^||^|y, B>.))
20		eleq1 1534	. . . . . . . 8 |- (x = |^||^|y -> (x e. A <-> |^||^|y e. A))
21	19, 20	anbi12d 628	. . . . . . 7 |- (x = |^||^|y -> ((y = <.x, B>. /\ x e. A) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
22	21	ceqsexgv 1888	. . . . . 6 |- (|^||^|y e. V -> (E.x(x = |^||^|y /\ (y = <.x, B>. /\ x e. A)) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
23		ancom 435	. . . . . . . . . . 11 |- (((y = <.x, z>. /\ x e. A) /\ z e. {B}) <-> (z e. {B} /\ (y = <.x, z>. /\ x e. A)))
24		anass 439	. . . . . . . . . . 11 |- (((y = <.x, z>. /\ x e. A) /\ z e. {B}) <-> (y = <.x, z>. /\ (x e. A /\ z e. {B})))
25		elsn 2421	. . . . . . . . . . . 12 |- (z e. {B} <-> z = B)
26	25	anbi1i 481	. . . . . . . . . . 11 |- ((z e. {B} /\ (y = <.x, z>. /\ x e. A)) <-> (z = B /\ (y = <.x, z>. /\ x e. A)))
27	23, 24, 26	3bitr3 181	. . . . . . . . . 10 |- ((y = <.x, z>. /\ (x e. A /\ z e. {B})) <-> (z = B /\ (y = <.x, z>. /\ x e. A)))
28	27	exbii 1051	. . . . . . . . 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 2488	. . . . . . . . . . . 12 |- (z = B -> <.x, z>. = <.x, B>.)
31	30	eqeq2d 1486	. . . . . . . . . . 11 |- (z = B -> (y = <.x, z>. <-> y = <.x, B>.))
32	31	anbi1d 617	. . . . . . . . . 10 |- (z = B -> ((y = <.x, z>. /\ x e. A) <-> (y = <.x, B>. /\ x e. A)))
33	29, 32	ceqsexv 1835	. . . . . . . . 9 |- (E.z(z = B /\ (y = <.x, z>. /\ x e. A)) <-> (y = <.x, B>. /\ x e. A))
34		inteq 2536	. . . . . . . . . . . . . 14 |- (y = <.x, B>. -> |^|y = |^|<.x, B>.)
35	34	inteqd 2538	. . . . . . . . . . . . 13 |- (y = <.x, B>. -> |^||^|y = |^||^|<.x, B>.)
36	7	op1stb 2913	. . . . . . . . . . . . 13 |- |^||^|<.x, B>. = x
37	35, 36	syl6req 1524	. . . . . . . . . . . 12 |- (y = <.x, B>. -> x = |^||^|y)
38	37	pm4.71ri 638	. . . . . . . . . . 11 |- (y = <.x, B>. <-> (x = |^||^|y /\ y = <.x, B>.))
39	38	anbi1i 481	. . . . . . . . . 10 |- ((y = <.x, B>. /\ x e. A) <-> ((x = |^||^|y /\ y = <.x, B>.) /\ x e. A))
40		anass 439	. . . . . . . . . 10 |- (((x = |^||^|y /\ y = <.x, B>.) /\ x e. A) <-> (x = |^||^|y /\ (y = <.x, B>. /\ x e. A)))
41	39, 40	bitr 173	. . . . . . . . 9 |- ((y = <.x, B>. /\ x e. A) <-> (x = |^||^|y /\ (y = <.x, B>. /\ x e. A)))
42	28, 33, 41	3bitr 177	. . . . . . . 8 |- (E.z(y = <.x, z>. /\ (x e. A /\ z e. {B})) <-> (x = |^||^|y /\ (y = <.x, B>. /\ x e. A)))
43	42	exbii 1051	. . . . . . 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	bitr 173	. . . . . 6 |- (y e. (A X. {B}) <-> E.x(x = |^||^|y /\ (y = <.x, B>. /\ x e. A)))
45	22, 44	syl5bb 532	. . . . 5 |- (|^||^|y e. V -> (y e. (A X. {B}) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
46	17, 45	syl 10	. . . 4 |- (x = |^||^|y -> (y e. (A X. {B}) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
47	46	pm5.32ri 646	. . 3 |- ((y e. (A X. {B}) /\ x = |^||^|y) <-> ((y = <.|^||^|y, B>. /\ |^||^|y e. A) /\ x = |^||^|y))
48	37	adantr 389	. . . . 5 |- ((y = <.x, B>. /\ x e. A) -> x = |^||^|y)
49	48	pm4.71i 637	. . . 4 |- ((y = <.x, B>. /\ x e. A) <-> ((y = <.x, B>. /\ x e. A) /\ x = |^||^|y))
50	21	pm5.32ri 646	. . . 4 |- (((y = <.x, B>. /\ x e. A) /\ x = |^||^|y) <-> ((y = <.|^||^|y, B>. /\ |^||^|y e. A) /\ x = |^||^|y))
51	49, 50	bitr2 174	. . 3 |- (((y = <.|^||^|y, B>. /\ |^||^|y e. A) /\ x = |^||^|y) <-> (y = <.x, B>. /\ x e. A))
52		ancom 435	. . 3 |- ((y = <.x, B>. /\ x e. A) <-> (x e. A /\ y = <.x, B>.))
53	47, 51, 52	3bitr 177	. 2 |- ((y e. (A X. {B}) /\ x = |^||^|y) <-> (x e. A /\ y = <.x, B>.))
54	3, 13, 15, 53	en2 4402	1 |- (A X. {B}) ~~ A

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811  {csn 2409  <.cop 2411  |^|cint 2533   class class class wbr 2619   X. cxp 3168   ~~ cen 4364

This theorem is referenced by:  xpsneng 4436  endisj 4437  xpdom3 4445  unxpdom2 4845  sucxpdom 4846  uncdadom 4921  cdaun 4922  pm110.643 4923  cdaen 4924  cda0en 4925  cda1en 4926  xp1en 4927  cdacomen 4929  cdaassen 4930  mapcdaen 4932  cdadom1 4933  xpnnen 7499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-en 4368

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