Theorem infmap2 7523

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. We start with infmap2lem2 7522 and also prove the other direction of the dominance relation. We obtain equinumerosity with Schroeder-Bernstein sbth 4437 and finally eliminate the degenerate case B = (/).

Hypotheses

Ref	Expression
infmap2.1	|- A e. V
infmap2.2	|- B e. V

Assertion

Ref	Expression
infmap2	|- ((om ~<_ A /\ B ~<_ A) -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})

Distinct variable groups:   x,A   x,B

Proof of Theorem infmap2

Step	Hyp	Ref	Expression
1		opreq2 3954	. . 3 |- (B = (/) -> (A ^m B) = (A ^m (/)))
2		breq2 2613	. . . . 5 |- (B = (/) -> (x ~~ B <-> x ~~ (/)))
3	2	anbi2d 614	. . . 4 |- (B = (/) -> ((x (_ A /\ x ~~ B) <-> (x (_ A /\ x ~~ (/))))
4	3	abbidv 1569	. . 3 |- (B = (/) -> {x | (x (_ A /\ x ~~ B)} = {x | (x (_ A /\ x ~~ (/))})
5	1, 4	breq12d 2621	. 2 |- (B = (/) -> ((A ^m B) ~~ {x | (x (_ A /\ x ~~ B)} <-> (A ^m (/)) ~~ {x | (x (_ A /\ x ~~ (/))}))
6		infmap2.1	. . . . . . . 8 |- A e. V
7		infmap2.2	. . . . . . . 8 |- B e. V
8	6, 7	infxpabs 7513	. . . . . . 7 |- ((om ~<_ A /\ B =/= (/) /\ B ~<_ A) -> (A X. B) ~~ A)
9	8	3com23 837	. . . . . 6 |- ((om ~<_ A /\ B ~<_ A /\ B =/= (/)) -> (A X. B) ~~ A)
10	9	3expa 831	. . . . 5 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (A X. B) ~~ A)
11	7, 6	xpcomen 4419	. . . . . 6 |- (B X. A) ~~ (A X. B)
12		entrt 4395	. . . . . 6 |- (((B X. A) ~~ (A X. B) /\ (A X. B) ~~ A) -> (B X. A) ~~ A)
13	11, 12	mpan 693	. . . . 5 |- ((A X. B) ~~ A -> (B X. A) ~~ A)
14	10, 13	syl 10	. . . 4 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (B X. A) ~~ A)
15	7, 6	xpex 3250	. . . . 5 |- (B X. A) e. V
16	15, 6	ssenen 4484	. . . 4 |- ((B X. A) ~~ A -> {x | (x (_ (B X. A) /\ x ~~ B)} ~~ {x | (x (_ A /\ x ~~ B)})
17		oprex 3968	. . . . . 6 |- (A ^m B) e. V
18		abid2 1572	. . . . . . 7 |- {x | x e. (A ^m B)} = (A ^m B)
19	6, 7	elmap 4318	. . . . . . . . 9 |- (x e. (A ^m B) <-> x:B-->A)
20		fssxp 3622	. . . . . . . . . 10 |- (x:B-->A -> x (_ (B X. A))
21		ffun 3615	. . . . . . . . . . . 12 |- (x:B-->A -> Fun x)
22		visset 1804	. . . . . . . . . . . . 13 |- x e. V
23	22	fundmen 4409	. . . . . . . . . . . 12 |- (Fun x -> dom x ~~ x)
24	22	ensym 4393	. . . . . . . . . . . 12 |- (dom x ~~ x -> x ~~ dom x)
25	21, 23, 24	3syl 20	. . . . . . . . . . 11 |- (x:B-->A -> x ~~ dom x)
26		fdm 3617	. . . . . . . . . . 11 |- (x:B-->A -> dom x = B)
27	25, 26	breqtrd 2629	. . . . . . . . . 10 |- (x:B-->A -> x ~~ B)
28	20, 27	jca 288	. . . . . . . . 9 |- (x:B-->A -> (x (_ (B X. A) /\ x ~~ B))
29	19, 28	sylbi 199	. . . . . . . 8 |- (x e. (A ^m B) -> (x (_ (B X. A) /\ x ~~ B))
30	29	ss2abi 2110	. . . . . . 7 |- {x | x e. (A ^m B)} (_ {x | (x (_ (B X. A) /\ x ~~ B)}
31	18, 30	eqsstr3 2082	. . . . . 6 |- (A ^m B) (_ {x | (x (_ (B X. A) /\ x ~~ B)}
32		ssdomg 4389	. . . . . 6 |- ((A ^m B) e. V -> ((A ^m B) (_ {x | (x (_ (B X. A) /\ x ~~ B)} -> (A ^m B) ~<_ {x | (x (_ (B X. A) /\ x ~~ B)}))
33	17, 31, 32	mp2 43	. . . . 5 |- (A ^m B) ~<_ {x | (x (_ (B X. A) /\ x ~~ B)}
34		domentr 4402	. . . . 5 |- (((A ^m B) ~<_ {x | (x (_ (B X. A) /\ x ~~ B)} /\ {x | (x (_ (B X. A) /\ x ~~ B)} ~~ {x | (x (_ A /\ x ~~ B)}) -> (A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)})
35	33, 34	mpan 693	. . . 4 |- ({x | (x (_ (B X. A) /\ x ~~ B)} ~~ {x | (x (_ A /\ x ~~ B)} -> (A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)})
36	14, 16, 35	3syl 20	. . 3 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)})
37		eqid 1468	. . . . 5 |- {<.z, w>. | ((z (_ A /\ z ~~ B) /\ w:B-onto->z)} = {<.z, w>. | ((z (_ A /\ z ~~ B) /\ w:B-onto->z)}
38	6, 7, 37	infmap2lem2 7522	. . . 4 |- {x | (x (_ A /\ x ~~ B)} ~<_ (A ^m B)
39		sbth 4437	. . . 4 |- (((A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)} /\ {x | (x (_ A /\ x ~~ B)} ~<_ (A ^m B)) -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})
40	38, 39	mpan2 694	. . 3 |- ((A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)} -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})
41	36, 40	syl 10	. 2 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})
42		1onn 4237	. . . . . 6 |- 1o e. om
43	42	elisseti 1809	. . . . 5 |- 1o e. V
44	43	enref 4372	. . . 4 |- 1o ~~ 1o
45	6	map0e 4326	. . . 4 |- (A ^m (/)) = 1o
46		df-sn 2402	. . . . 5 |- {(/)} = {x | x = (/)}
47		df1o2 4124	. . . . 5 |- 1o = {(/)}
48		en0 4404	. . . . . . . 8 |- (x ~~ (/) <-> x = (/))
49	48	anbi2i 479	. . . . . . 7 |- ((x (_ A /\ x ~~ (/)) <-> (x (_ A /\ x = (/)))
50		0ss 2291	. . . . . . . . 9 |- (/) (_ A
51		sseq1 2072	. . . . . . . . 9 |- (x = (/) -> (x (_ A <-> (/) (_ A))
52	50, 51	mpbiri 194	. . . . . . . 8 |- (x = (/) -> x (_ A)
53	52	pm4.71ri 636	. . . . . . 7 |- (x = (/) <-> (x (_ A /\ x = (/)))
54	49, 53	bitr4 176	. . . . . 6 |- ((x (_ A /\ x ~~ (/)) <-> x = (/))
55	54	abbii 1567	. . . . 5 |- {x | (x (_ A /\ x ~~ (/))} = {x | x = (/)}
56	46, 47, 55	3eqtr4r 1498	. . . 4 |- {x | (x (_ A /\ x ~~ (/))} = 1o
57	44, 45, 56	3brtr4 2633	. . 3 |- (A ^m (/)) ~~ {x | (x (_ A /\ x ~~ (/))}
58	57	a1i 8	. 2 |- ((om ~<_ A /\ B ~<_ A) -> (A ^m (/)) ~~ {x | (x (_ A /\ x ~~ (/))})
59	5, 41, 58	pm2.61ne 1625	1 |- ((om ~<_ A /\ B ~<_ A) -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456   =/= wne 1577  Vcvv 1802   (_ wss 2037  (/)c0 2270  {csn 2399   class class class wbr 2609  {copab 2656  omcom 3121   X. cxp 3158  dom cdm 3160  Fun wfun 3166  -->wf 3168  -onto->wfo 3170  (class class class)co 3948  1oc1o 4112   ^m cm 4306   ~~ cen 4348   ~<_ cdom 4349

This theorem is referenced by:  alephexp2 7528

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-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  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-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  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-fv 3188  df-iso 3189  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-2o 4118  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-map 4308  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-n 5873  df-2 5917  df-n0 6047  df-z 6083  df-seq1 6245  df-exp 6501

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