Theorem xpsnen 4415

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 2740	. . 3 |- {B} e. V
3	1, 2	xpex 3250	. 2 |- (A X. {B}) e. V
4		elxp 3192	. . 3 |- (y e. (A X. {B}) <-> E.xE.z(y = <.x, z>. /\ (x e. A /\ z e. {B})))
5		inteq 2526	. . . . . . . 8 |- (y = <.x, z>. -> |^|y = |^|<.x, z>.)
6	5	inteqd 2528	. . . . . . 7 |- (y = <.x, z>. -> |^||^|y = |^||^|<.x, z>.)
7		visset 1804	. . . . . . . 8 |- x e. V
8	7	op1stb 2903	. . . . . . 7 |- |^||^|<.x, z>. = x
9	6, 8	syl6eq 1515	. . . . . 6 |- (y = <.x, z>. -> |^||^|y = x)
10	9, 7	syl6eqel 1548	. . . . 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 1291	. . 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 2772	. . 3 |- <.x, B>. e. V
15	14	a1i 8	. 2 |- (x e. A -> <.x, B>. e. V)
16		eleq1 1526	. . . . . 6 |- (x = |^||^|y -> (x e. V <-> |^||^|y e. V))
17	7, 16	mpbii 193	. . . . 5 |- (x = |^||^|y -> |^||^|y e. V)
18		opeq1 2478	. . . . . . . . 9 |- (x = |^||^|y -> <.x, B>. = <.|^||^|y, B>.)
19	18	eqeq2d 1478	. . . . . . . 8 |- (x = |^||^|y -> (y = <.x, B>. <-> y = <.|^||^|y, B>.))
20		eleq1 1526	. . . . . . . 8 |- (x = |^||^|y -> (x e. A <-> |^||^|y e. A))
21	19, 20	anbi12d 626	. . . . . . 7 |- (x = |^||^|y -> ((y = <.x, B>. /\ x e. A) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
22	21	ceqsexgv 1879	. . . . . 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 2411	. . . . . . . . . . . 12 |- (z e. {B} <-> z = B)
26	25	anbi1i 480	. . . . . . . . . . 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 1047	. . . . . . . . 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 2479	. . . . . . . . . . . 12 |- (z = B -> <.x, z>. = <.x, B>.)
31	30	eqeq2d 1478	. . . . . . . . . . 11 |- (z = B -> (y = <.x, z>. <-> y = <.x, B>.))
32	31	anbi1d 615	. . . . . . . . . 10 |- (z = B -> ((y = <.x, z>. /\ x e. A) <-> (y = <.x, B>. /\ x e. A)))
33	29, 32	ceqsexv 1826	. . . . . . . . 9 |- (E.z(z = B /\ (y = <.x, z>. /\ x e. A)) <-> (y = <.x, B>. /\ x e. A))
34		inteq 2526	. . . . . . . . . . . . . 14 |- (y = <.x, B>. -> |^|y = |^|<.x, B>.)
35	34	inteqd 2528	. . . . . . . . . . . . 13 |- (y = <.x, B>. -> |^||^|y = |^||^|<.x, B>.)
36	7	op1stb 2903	. . . . . . . . . . . . 13 |- |^||^|<.x, B>. = x
37	35, 36	syl6req 1516	. . . . . . . . . . . 12 |- (y = <.x, B>. -> x = |^||^|y)
38	37	pm4.71ri 636	. . . . . . . . . . 11 |- (y = <.x, B>. <-> (x = |^||^|y /\ y = <.x, B>.))
39	38	anbi1i 480	. . . . . . . . . 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 1047	. . . . . . 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 530	. . . . 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 644	. . 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 635	. . . 4 |- ((y = <.x, B>. /\ x e. A) <-> ((y = <.x, B>. /\ x e. A) /\ x = |^||^|y))
50	21	pm5.32ri 644	. . . 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 4383	1 |- (A X. {B}) ~~ A

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802  {csn 2399  <.cop 2401  |^|cint 2523   class class class wbr 2609   X. cxp 3158   ~~ cen 4348

This theorem is referenced by:  xpsneng 4416  endisj 4417  xpdom3 4425  unxpdom2 4817  sucxpdom 4818  uncdadom 4893  cdaun 4894  pm110.643 4895  cdaen 4896  cda0en 4897  cda1en 4898  xp1en 4899  cdacomen 4901  cdaassen 4902  mapcdaen 4904  cdadom1 4905  xpnnen 7441

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-en 4351

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