Theorem infmap2 8850

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. We start with infmap2lem2 8849 and also prove the other direction of the dominance relation. We obtain equinumerosity with Schroeder-Bernstein sbth 5520 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 C_ A /\ x ~~ B)})

Distinct variable groups:   x,A   x,B

Proof of Theorem infmap2

Step	Hyp	Ref	Expression
1		opreq2 4890	. . 3 |- (B = (/) -> (A ^m B) = (A ^m (/)))
2		breq2 3342	. . . . 5 |- (B = (/) -> (x ~~ B <-> x ~~ (/)))
3	2	anbi2d 678	. . . 4 |- (B = (/) -> ((x C_ A /\ x ~~ B) <-> (x C_ A /\ x ~~ (/))))
4	3	abbidv 2008	. . 3 |- (B = (/) -> {x | (x C_ A /\ x ~~ B)} = {x | (x C_ A /\ x ~~ (/))})
5	1, 4	breq12d 3351	. 2 |- (B = (/) -> ((A ^m B) ~~ {x | (x C_ A /\ x ~~ B)} <-> (A ^m (/)) ~~ {x | (x C_ A /\ x ~~ (/))}))
6		sbth 5520	. . 3 |- (((A ^m B) ~<_ {x | (x C_ A /\ x ~~ B)} /\ {x | (x C_ A /\ x ~~ B)} ~<_ (A ^m B)) -> (A ^m B) ~~ {x | (x C_ A /\ x ~~ B)})
7		entr 5473	. . . . 5 |- (((B X. A) ~~ (A X. B) /\ (A X. B) ~~ A) -> (B X. A) ~~ A)
8		infmap2.2	. . . . . 6 |- B e. _V
9		infmap2.1	. . . . . 6 |- A e. _V
10	8, 9	xpcomen 5498	. . . . 5 |- (B X. A) ~~ (A X. B)
11	9, 8	infxpabs 8839	. . . . . . 7 |- ((om ~<_ A /\ B =/= (/) /\ B ~<_ A) -> (A X. B) ~~ A)
12	11	3com23 1074	. . . . . 6 |- ((om ~<_ A /\ B ~<_ A /\ B =/= (/)) -> (A X. B) ~~ A)
13	12	3expa 1067	. . . . 5 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (A X. B) ~~ A)
14	7, 10, 13	sylancr 526	. . . 4 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (B X. A) ~~ A)
15	8, 9	xpex 4096	. . . . 5 |- (B X. A) e. _V
16	15, 9	ssenen 5598	. . . 4 |- ((B X. A) ~~ A -> {x | (x C_ (B X. A) /\ x ~~ B)} ~~ {x | (x C_ A /\ x ~~ B)})
17		oprex 4907	. . . . . 6 |- (A ^m B) e. _V
18		abid2 2011	. . . . . . 7 |- {x | x e. (A ^m B)} = (A ^m B)
19	9, 8	elmap 5393	. . . . . . . . 9 |- (x e. (A ^m B) <-> x:B-->A)
20		fssxp 4575	. . . . . . . . . 10 |- (x:B-->A -> x C_ (B X. A))
21		ffun 4565	. . . . . . . . . . . 12 |- (x:B-->A -> Fun x)
22		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
23	22	fundmen 5487	. . . . . . . . . . . 12 |- (Fun x -> dom x ~~ x)
24	22	ensym 5471	. . . . . . . . . . . 12 |- (dom x ~~ x -> x ~~ dom x)
25	21, 23, 24	3syl 24	. . . . . . . . . . 11 |- (x:B-->A -> x ~~ dom x)
26		fdm 4567	. . . . . . . . . . 11 |- (x:B-->A -> dom x = B)
27	25, 26	breqtrd 3361	. . . . . . . . . 10 |- (x:B-->A -> x ~~ B)
28	20, 27	jca 310	. . . . . . . . 9 |- (x:B-->A -> (x C_ (B X. A) /\ x ~~ B))
29	19, 28	sylbi 216	. . . . . . . 8 |- (x e. (A ^m B) -> (x C_ (B X. A) /\ x ~~ B))
30	29	ss2abi 2679	. . . . . . 7 |- {x | x e. (A ^m B)} C_ {x | (x C_ (B X. A) /\ x ~~ B)}
31	18, 30	eqsstr3i 2648	. . . . . 6 |- (A ^m B) C_ {x | (x C_ (B X. A) /\ x ~~ B)}
32		ssdomg 5467	. . . . . 6 |- ((A ^m B) e. _V -> ((A ^m B) C_ {x | (x C_ (B X. A) /\ x ~~ B)} -> (A ^m B) ~<_ {x | (x C_ (B X. A) /\ x ~~ B)}))
33	17, 31, 32	mp2 54	. . . . 5 |- (A ^m B) ~<_ {x | (x C_ (B X. A) /\ x ~~ B)}
34		domentr 5480	. . . . 5 |- (((A ^m B) ~<_ {x | (x C_ (B X. A) /\ x ~~ B)} /\ {x | (x C_ (B X. A) /\ x ~~ B)} ~~ {x | (x C_ A /\ x ~~ B)}) -> (A ^m B) ~<_ {x | (x C_ A /\ x ~~ B)})
35	33, 34	mpan 759	. . . 4 |- ({x | (x C_ (B X. A) /\ x ~~ B)} ~~ {x | (x C_ A /\ x ~~ B)} -> (A ^m B) ~<_ {x | (x C_ A /\ x ~~ B)})
36	14, 16, 35	3syl 24	. . 3 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (A ^m B) ~<_ {x | (x C_ A /\ x ~~ B)})
37		eqid 1884	. . . 4 |- {<.z, w>. | ((z C_ A /\ z ~~ B) /\ w:B-onto->z)} = {<.z, w>. | ((z C_ A /\ z ~~ B) /\ w:B-onto->z)}
38	9, 8, 37	infmap2lem2 8849	. . 3 |- {x | (x C_ A /\ x ~~ B)} ~<_ (A ^m B)
39	6, 36, 38	sylancl 525	. 2 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (A ^m B) ~~ {x | (x C_ A /\ x ~~ B)})
40		1onn 5310	. . . . . 6 |- 1o e. om
41	40	elisseti 2301	. . . . 5 |- 1o e. _V
42	41	enref 5450	. . . 4 |- 1o ~~ 1o
43	9	map0e 5401	. . . 4 |- (A ^m (/)) = 1o
44		df-sn 3049	. . . . 5 |- {(/)} = {x | x = (/)}
45		df1o2 5185	. . . . 5 |- 1o = {(/)}
46		en0 5482	. . . . . . . 8 |- (x ~~ (/) <-> x = (/))
47	46	anbi2i 538	. . . . . . 7 |- ((x C_ A /\ x ~~ (/)) <-> (x C_ A /\ x = (/)))
48		0ss 2900	. . . . . . . . 9 |- (/) C_ A
49		sseq1 2637	. . . . . . . . 9 |- (x = (/) -> (x C_ A <-> (/) C_ A))
50	48, 49	mpbiri 211	. . . . . . . 8 |- (x = (/) -> x C_ A)
51	50	pm4.71ri 700	. . . . . . 7 |- (x = (/) <-> (x C_ A /\ x = (/)))
52	47, 51	bitr4i 193	. . . . . 6 |- ((x C_ A /\ x ~~ (/)) <-> x = (/))
53	52	abbii 2006	. . . . 5 |- {x | (x C_ A /\ x ~~ (/))} = {x | x = (/)}
54	44, 45, 53	3eqtr4ri 1923	. . . 4 |- {x | (x C_ A /\ x ~~ (/))} = 1o
55	42, 43, 54	3brtr4i 3365	. . 3 |- (A ^m (/)) ~~ {x | (x C_ A /\ x ~~ (/))}
56	55	a1i 8	. 2 |- ((om ~<_ A /\ B ~<_ A) -> (A ^m (/)) ~~ {x | (x C_ A /\ x ~~ (/))})
57	5, 39, 56	pm2.61ne 2087	1 |- ((om ~<_ A /\ B ~<_ A) -> (A ^m B) ~~ {x | (x C_ A /\ x ~~ B)})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044   class class class wbr 3338  {copab 3395  omcom 3949   X. cxp 3984  dom cdm 3986  Fun wfun 3992  -->wf 3994  -onto->wfo 3996  (class class class)co 4884  1oc1o 5172   ^m cmap 5381   ~~ cen 5423   ~<_ cdom 5424

This theorem is referenced by:  alephexp2 8855

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  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-fv 4014  df-iso 4015  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-2o 5178  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-undef 5556  df-riota 5560  df-card 5862  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

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