Theorem infmap2 9040

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9398 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem infmap2

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . 3 |- ( B = (/) -> ( A ^m B ) = ( A ^m (/) ) )
2		breq2 4657	. . . . 5 |- ( B = (/) -> ( x ~~ B <-> x ~~ (/) ) )
3	2	anbi2d 740	. . . 4 |- ( B = (/) -> ( ( x C_ A /\ x ~~ B ) <-> ( x C_ A /\ x ~~ (/) ) ) )
4	3	abbidv 2741	. . 3 |- ( B = (/) -> { x | ( x C_ A /\ x ~~ B ) } = { x | ( x C_ A /\ x ~~ (/) ) } )
5	1, 4	breq12d 4666	. 2 |- ( B = (/) -> ( ( A ^m B ) ~~ { x | ( x C_ A /\ x ~~ B ) } <-> ( A ^m (/) ) ~~ { x | ( x C_ A /\ x ~~ (/) ) } ) )
6		simpl2 1065	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> B ~<_ A )
7		reldom 7961	. . . . . . . . . . 11 |- Rel ~<_
8	7	brrelexi 5158	. . . . . . . . . 10 |- ( B ~<_ A -> B e. _V )
9	6, 8	syl 17	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> B e. _V )
10	7	brrelex2i 5159	. . . . . . . . . 10 |- ( B ~<_ A -> A e. _V )
11	6, 10	syl 17	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> A e. _V )
12		xpcomeng 8052	. . . . . . . . 9 |- ( ( B e. _V /\ A e. _V ) -> ( B X. A ) ~~ ( A X. B ) )
13	9, 11, 12	syl2anc 693	. . . . . . . 8 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( B X. A ) ~~ ( A X. B ) )
14		simpl3 1066	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( A ^m B ) e. dom card )
15		simpr 477	. . . . . . . . . . 11 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> B =/= (/) )
16		mapdom3 8132	. . . . . . . . . . 11 |- ( ( A e. _V /\ B e. _V /\ B =/= (/) ) -> A ~<_ ( A ^m B ) )
17	11, 9, 15, 16	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> A ~<_ ( A ^m B ) )
18		numdom 8861	. . . . . . . . . 10 |- ( ( ( A ^m B ) e. dom card /\ A ~<_ ( A ^m B ) ) -> A e. dom card )
19	14, 17, 18	syl2anc 693	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> A e. dom card )
20		simpl1 1064	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> om ~<_ A )
21		infxpabs 9034	. . . . . . . . 9 |- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B =/= (/) /\ B ~<_ A ) ) -> ( A X. B ) ~~ A )
22	19, 20, 15, 6, 21	syl22anc 1327	. . . . . . . 8 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( A X. B ) ~~ A )
23		entr 8008	. . . . . . . 8 |- ( ( ( B X. A ) ~~ ( A X. B ) /\ ( A X. B ) ~~ A ) -> ( B X. A ) ~~ A )
24	13, 22, 23	syl2anc 693	. . . . . . 7 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( B X. A ) ~~ A )
25		ssenen 8134	. . . . . . 7 |- ( ( B X. A ) ~~ A -> { x | ( x C_ ( B X. A ) /\ x ~~ B ) } ~~ { x | ( x C_ A /\ x ~~ B ) } )
26	24, 25	syl 17	. . . . . 6 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> { x | ( x C_ ( B X. A ) /\ x ~~ B ) } ~~ { x | ( x C_ A /\ x ~~ B ) } )
27		relen 7960	. . . . . . 7 |- Rel ~~
28	27	brrelexi 5158	. . . . . 6 |- ( { x | ( x C_ ( B X. A ) /\ x ~~ B ) } ~~ { x | ( x C_ A /\ x ~~ B ) } -> { x | ( x C_ ( B X. A ) /\ x ~~ B ) } e. _V )
29	26, 28	syl 17	. . . . 5 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> { x | ( x C_ ( B X. A ) /\ x ~~ B ) } e. _V )
30		abid2 2745	. . . . . 6 |- { x | x e. ( A ^m B ) } = ( A ^m B )
31		elmapi 7879	. . . . . . . 8 |- ( x e. ( A ^m B ) -> x : B --> A )
32		fssxp 6060	. . . . . . . . 9 |- ( x : B --> A -> x C_ ( B X. A ) )
33		ffun 6048	. . . . . . . . . . 11 |- ( x : B --> A -> Fun x )
34		vex 3203	. . . . . . . . . . . 12 |- x e. _V
35	34	fundmen 8030	. . . . . . . . . . 11 |- ( Fun x -> dom x ~~ x )
36		ensym 8005	. . . . . . . . . . 11 |- ( dom x ~~ x -> x ~~ dom x )
37	33, 35, 36	3syl 18	. . . . . . . . . 10 |- ( x : B --> A -> x ~~ dom x )
38		fdm 6051	. . . . . . . . . 10 |- ( x : B --> A -> dom x = B )
39	37, 38	breqtrd 4679	. . . . . . . . 9 |- ( x : B --> A -> x ~~ B )
40	32, 39	jca 554	. . . . . . . 8 |- ( x : B --> A -> ( x C_ ( B X. A ) /\ x ~~ B ) )
41	31, 40	syl 17	. . . . . . 7 |- ( x e. ( A ^m B ) -> ( x C_ ( B X. A ) /\ x ~~ B ) )
42	41	ss2abi 3674	. . . . . 6 |- { x | x e. ( A ^m B ) } C_ { x | ( x C_ ( B X. A ) /\ x ~~ B ) }
43	30, 42	eqsstr3i 3636	. . . . 5 |- ( A ^m B ) C_ { x | ( x C_ ( B X. A ) /\ x ~~ B ) }
44		ssdomg 8001	. . . . 5 |- ( { x | ( x C_ ( B X. A ) /\ x ~~ 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 ) } ) )
45	29, 43, 44	mpisyl 21	. . . 4 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( A ^m B ) ~<_ { x | ( x C_ ( B X. A ) /\ x ~~ B ) } )
46		domentr 8015	. . . 4 |- ( ( ( 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 ) } )
47	45, 26, 46	syl2anc 693	. . 3 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( A ^m B ) ~<_ { x | ( x C_ A /\ x ~~ B ) } )
48		ovex 6678	. . . . . . 7 |- ( A ^m B ) e. _V
49	48	mptex 6486	. . . . . 6 |- ( f e. ( A ^m B ) |-> ran f ) e. _V
50	49	rnex 7100	. . . . 5 |- ran ( f e. ( A ^m B ) |-> ran f ) e. _V
51		ensym 8005	. . . . . . . . . . . 12 |- ( x ~~ B -> B ~~ x )
52	51	ad2antll 765	. . . . . . . . . . 11 |- ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) -> B ~~ x )
53		bren 7964	. . . . . . . . . . 11 |- ( B ~~ x <-> E. f f : B -1-1-onto-> x )
54	52, 53	sylib 208	. . . . . . . . . 10 |- ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) -> E. f f : B -1-1-onto-> x )
55		f1of 6137	. . . . . . . . . . . . . . . 16 |- ( f : B -1-1-onto-> x -> f : B --> x )
56	55	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) /\ f : B -1-1-onto-> x ) -> f : B --> x )
57		simplrl 800	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) /\ f : B -1-1-onto-> x ) -> x C_ A )
58	56, 57	fssd 6057	. . . . . . . . . . . . . 14 |- ( ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) /\ f : B -1-1-onto-> x ) -> f : B --> A )
59	11, 9	elmapd 7871	. . . . . . . . . . . . . . 15 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( f e. ( A ^m B ) <-> f : B --> A ) )
60	59	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) /\ f : B -1-1-onto-> x ) -> ( f e. ( A ^m B ) <-> f : B --> A ) )
61	58, 60	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) /\ f : B -1-1-onto-> x ) -> f e. ( A ^m B ) )
62		f1ofo 6144	. . . . . . . . . . . . . . . 16 |- ( f : B -1-1-onto-> x -> f : B -onto-> x )
63		forn 6118	. . . . . . . . . . . . . . . 16 |- ( f : B -onto-> x -> ran f = x )
64	62, 63	syl 17	. . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> x -> ran f = x )
65	64	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) /\ f : B -1-1-onto-> x ) -> ran f = x )
66	65	eqcomd 2628	. . . . . . . . . . . . 13 |- ( ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) /\ f : B -1-1-onto-> x ) -> x = ran f )
67	61, 66	jca 554	. . . . . . . . . . . 12 |- ( ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) /\ f : B -1-1-onto-> x ) -> ( f e. ( A ^m B ) /\ x = ran f ) )
68	67	ex 450	. . . . . . . . . . 11 |- ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) -> ( f : B -1-1-onto-> x -> ( f e. ( A ^m B ) /\ x = ran f ) ) )
69	68	eximdv 1846	. . . . . . . . . 10 |- ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) -> ( E. f f : B -1-1-onto-> x -> E. f ( f e. ( A ^m B ) /\ x = ran f ) ) )
70	54, 69	mpd 15	. . . . . . . . 9 |- ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) -> E. f ( f e. ( A ^m B ) /\ x = ran f ) )
71		df-rex 2918	. . . . . . . . 9 |- ( E. f e. ( A ^m B ) x = ran f <-> E. f ( f e. ( A ^m B ) /\ x = ran f ) )
72	70, 71	sylibr 224	. . . . . . . 8 |- ( ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) /\ ( x C_ A /\ x ~~ B ) ) -> E. f e. ( A ^m B ) x = ran f )
73	72	ex 450	. . . . . . 7 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( ( x C_ A /\ x ~~ B ) -> E. f e. ( A ^m B ) x = ran f ) )
74	73	ss2abdv 3675	. . . . . 6 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> { x | ( x C_ A /\ x ~~ B ) } C_ { x | E. f e. ( A ^m B ) x = ran f } )
75		eqid 2622	. . . . . . 7 |- ( f e. ( A ^m B ) |-> ran f ) = ( f e. ( A ^m B ) |-> ran f )
76	75	rnmpt 5371	. . . . . 6 |- ran ( f e. ( A ^m B ) |-> ran f ) = { x | E. f e. ( A ^m B ) x = ran f }
77	74, 76	syl6sseqr 3652	. . . . 5 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> { x | ( x C_ A /\ x ~~ B ) } C_ ran ( f e. ( A ^m B ) |-> ran f ) )
78		ssdomg 8001	. . . . 5 |- ( ran ( f e. ( A ^m B ) |-> ran f ) e. _V -> ( { x | ( x C_ A /\ x ~~ B ) } C_ ran ( f e. ( A ^m B ) |-> ran f ) -> { x | ( x C_ A /\ x ~~ B ) } ~<_ ran ( f e. ( A ^m B ) |-> ran f ) ) )
79	50, 77, 78	mpsyl 68	. . . 4 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> { x | ( x C_ A /\ x ~~ B ) } ~<_ ran ( f e. ( A ^m B ) |-> ran f ) )
80		vex 3203	. . . . . . . . 9 |- f e. _V
81	80	rnex 7100	. . . . . . . 8 |- ran f e. _V
82	81	rgenw 2924	. . . . . . 7 |- A. f e. ( A ^m B ) ran f e. _V
83	75	fnmpt 6020	. . . . . . 7 |- ( A. f e. ( A ^m B ) ran f e. _V -> ( f e. ( A ^m B ) |-> ran f ) Fn ( A ^m B ) )
84	82, 83	mp1i 13	. . . . . 6 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( f e. ( A ^m B ) |-> ran f ) Fn ( A ^m B ) )
85		dffn4 6121	. . . . . 6 |- ( ( f e. ( A ^m B ) |-> ran f ) Fn ( A ^m B ) <-> ( f e. ( A ^m B ) |-> ran f ) : ( A ^m B ) -onto-> ran ( f e. ( A ^m B ) |-> ran f ) )
86	84, 85	sylib 208	. . . . 5 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( f e. ( A ^m B ) |-> ran f ) : ( A ^m B ) -onto-> ran ( f e. ( A ^m B ) |-> ran f ) )
87		fodomnum 8880	. . . . 5 |- ( ( A ^m B ) e. dom card -> ( ( f e. ( A ^m B ) |-> ran f ) : ( A ^m B ) -onto-> ran ( f e. ( A ^m B ) |-> ran f ) -> ran ( f e. ( A ^m B ) |-> ran f ) ~<_ ( A ^m B ) ) )
88	14, 86, 87	sylc 65	. . . 4 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ran ( f e. ( A ^m B ) |-> ran f ) ~<_ ( A ^m B ) )
89		domtr 8009	. . . 4 |- ( ( { x | ( x C_ A /\ x ~~ B ) } ~<_ ran ( f e. ( A ^m B ) |-> ran f ) /\ ran ( f e. ( A ^m B ) |-> ran f ) ~<_ ( A ^m B ) ) -> { x | ( x C_ A /\ x ~~ B ) } ~<_ ( A ^m B ) )
90	79, 88, 89	syl2anc 693	. . 3 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> { x | ( x C_ A /\ x ~~ B ) } ~<_ ( A ^m B ) )
91		sbth 8080	. . 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 ) } )
92	47, 90, 91	syl2anc 693	. 2 |- ( ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) /\ B =/= (/) ) -> ( A ^m B ) ~~ { x | ( x C_ A /\ x ~~ B ) } )
93	7	brrelex2i 5159	. . . . 5 |- ( om ~<_ A -> A e. _V )
94	93	3ad2ant1 1082	. . . 4 |- ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) -> A e. _V )
95		map0e 7895	. . . 4 |- ( A e. _V -> ( A ^m (/) ) = 1o )
96	94, 95	syl 17	. . 3 |- ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) -> ( A ^m (/) ) = 1o )
97		1onn 7719	. . . . . 6 |- 1o e. om
98	97	elexi 3213	. . . . 5 |- 1o e. _V
99	98	enref 7988	. . . 4 |- 1o ~~ 1o
100		df-sn 4178	. . . . 5 |- { (/) } = { x | x = (/) }
101		df1o2 7572	. . . . 5 |- 1o = { (/) }
102		en0 8019	. . . . . . . 8 |- ( x ~~ (/) <-> x = (/) )
103	102	anbi2i 730	. . . . . . 7 |- ( ( x C_ A /\ x ~~ (/) ) <-> ( x C_ A /\ x = (/) ) )
104		0ss 3972	. . . . . . . . 9 |- (/) C_ A
105		sseq1 3626	. . . . . . . . 9 |- ( x = (/) -> ( x C_ A <-> (/) C_ A ) )
106	104, 105	mpbiri 248	. . . . . . . 8 |- ( x = (/) -> x C_ A )
107	106	pm4.71ri 665	. . . . . . 7 |- ( x = (/) <-> ( x C_ A /\ x = (/) ) )
108	103, 107	bitr4i 267	. . . . . 6 |- ( ( x C_ A /\ x ~~ (/) ) <-> x = (/) )
109	108	abbii 2739	. . . . 5 |- { x | ( x C_ A /\ x ~~ (/) ) } = { x | x = (/) }
110	100, 101, 109	3eqtr4ri 2655	. . . 4 |- { x | ( x C_ A /\ x ~~ (/) ) } = 1o
111	99, 110	breqtrri 4680	. . 3 |- 1o ~~ { x | ( x C_ A /\ x ~~ (/) ) }
112	96, 111	syl6eqbr 4692	. 2 |- ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) -> ( A ^m (/) ) ~~ { x | ( x C_ A /\ x ~~ (/) ) } )
113	5, 92, 112	pm2.61ne 2879	1 |- ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) -> ( A ^m B ) ~~ { x | ( x C_ A /\ x ~~ B ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887  (class class class)co 6650   omcom 7065   1oc1o 7553   ^m cmap 7857   ~~ cen 7952   ~<_ cdom 7953   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768

This theorem is referenced by:  infmap  9398