Theorem xpsnen 4424

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 2746	. . 3 |- {B} e. V
3	1, 2	xpex 3256	. 2 |- (A X. {B}) e. V
4		elxp 3198	. . 3 |- (y e. (A X. {B}) <-> E.xE.z(y = <.x, z>. /\ (x e. A /\ z e. {B})))
5		inteq 2532	. . . . . . . 8 |- (y = <.x, z>. -> |^|y = |^|<.x, z>.)
6	5	inteqd 2534	. . . . . . 7 |- (y = <.x, z>. -> |^||^|y = |^||^|<.x, z>.)
7		visset 1810	. . . . . . . 8 |- x e. V
8	7	op1stb 2909	. . . . . . 7 |- |^||^|<.x, z>. = x
9	6, 8	syl6eq 1521	. . . . . 6 |- (y = <.x, z>. -> |^||^|y = x)
10	9, 7	syl6eqel 1554	. . . . 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 1295	. . 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 2778	. . 3 |- <.x, B>. e. V
15	14	a1i 8	. 2 |- (x e. A -> <.x, B>. e. V)
16		eleq1 1532	. . . . . 6 |- (x = |^||^|y -> (x e. V <-> |^||^|y e. V))
17	7, 16	mpbii 193	. . . . 5 |- (x = |^||^|y -> |^||^|y e. V)
18		opeq1 2484	. . . . . . . . 9 |- (x = |^||^|y -> <.x, B>. = <.|^||^|y, B>.)
19	18	eqeq2d 1484	. . . . . . . 8 |- (x = |^||^|y -> (y = <.x, B>. <-> y = <.|^||^|y, B>.))
20		eleq1 1532	. . . . . . . 8 |- (x = |^||^|y -> (x e. A <-> |^||^|y e. A))
21	19, 20	anbi12d 627	. . . . . . 7 |- (x = |^||^|y -> ((y = <.x, B>. /\ x e. A) <-> (y = <.|^||^|y, B>. /\ |^||^|y e. A)))
22	21	ceqsexgv 1885	. . . . . 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 2418	. . . . . . . . . . . 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 1050	. . . . . . . . 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 2485	. . . . . . . . . . . 12 |- (z = B -> <.x, z>. = <.x, B>.)
31	30	eqeq2d 1484	. . . . . . . . . . 11 |- (z = B -> (y = <.x, z>. <-> y = <.x, B>.))
32	31	anbi1d 616	. . . . . . . . . 10 |- (z = B -> ((y = <.x, z>. /\ x e. A) <-> (y = <.x, B>. /\ x e. A)))
33	29, 32	ceqsexv 1832	. . . . . . . . 9 |- (E.z(z = B /\ (y = <.x, z>. /\ x e. A)) <-> (y = <.x, B>. /\ x e. A))
34		inteq 2532	. . . . . . . . . . . . . 14 |- (y = <.x, B>. -> |^|y = |^|<.x, B>.)
35	34	inteqd 2534	. . . . . . . . . . . . 13 |- (y = <.x, B>. -> |^||^|y = |^||^|<.x, B>.)
36	7	op1stb 2909	. . . . . . . . . . . . 13 |- |^||^|<.x, B>. = x
37	35, 36	syl6req 1522	. . . . . . . . . . . 12 |- (y = <.x, B>. -> x = |^||^|y)
38	37	pm4.71ri 637	. . . . . . . . . . 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 1050	. . . . . . 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 531	. . . . 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 645	. . 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 636	. . . 4 |- ((y = <.x, B>. /\ x e. A) <-> ((y = <.x, B>. /\ x e. A) /\ x = |^||^|y))
50	21	pm5.32ri 645	. . . 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 4392	1 |- (A X. {B}) ~~ A

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  Vcvv 1808  {csn 2406  <.cop 2408  |^|cint 2529   class class class wbr 2615   X. cxp 3164   ~~ cen 4357

This theorem is referenced by:  xpsneng 4425  endisj 4426  xpdom3 4434  unxpdom2 4828  sucxpdom 4829  uncdadom 4904  cdaun 4905  pm110.643 4906  cdaen 4907  cda0en 4908  cda1en 4909  xp1en 4910  cdacomen 4912  cdaassen 4913  mapcdaen 4915  cdadom1 4916  xpnnen 7458

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-int 2530  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-en 4360

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