Theorem r1om 9066

Description: The set of hereditarily finite sets is countable. See ackbij2 9065 for an explicit bijection that works without Infinity. See also r1omALT 9598. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Assertion

Ref	Expression
r1om	|- ( R1 ` om ) ~~ om

Proof of Theorem r1om

Dummy variables a b c d e f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omex 8540	. . . 4 |- om e. _V
2		limom 7080	. . . 4 |- Lim om
3		r1lim 8635	. . . 4 |- ( ( om e. _V /\ Lim om ) -> ( R1 ` om ) = U_ a e. om ( R1 ` a ) )
4	1, 2, 3	mp2an 708	. . 3 |- ( R1 ` om ) = U_ a e. om ( R1 ` a )
5		r1fnon 8630	. . . 4 |- R1 Fn On
6		fnfun 5988	. . . 4 |- ( R1 Fn On -> Fun R1 )
7		funiunfv 6506	. . . 4 |- ( Fun R1 -> U_ a e. om ( R1 ` a ) = U. ( R1 " om ) )
8	5, 6, 7	mp2b 10	. . 3 |- U_ a e. om ( R1 ` a ) = U. ( R1 " om )
9	4, 8	eqtri 2644	. 2 |- ( R1 ` om ) = U. ( R1 " om )
10		iuneq1 4534	. . . . . . 7 |- ( e = a -> U_ f e. e ( { f } X. ~P f ) = U_ f e. a ( { f } X. ~P f ) )
11		sneq 4187	. . . . . . . . 9 |- ( f = b -> { f } = { b } )
12		pweq 4161	. . . . . . . . 9 |- ( f = b -> ~P f = ~P b )
13	11, 12	xpeq12d 5140	. . . . . . . 8 |- ( f = b -> ( { f } X. ~P f ) = ( { b } X. ~P b ) )
14	13	cbviunv 4559	. . . . . . 7 |- U_ f e. a ( { f } X. ~P f ) = U_ b e. a ( { b } X. ~P b )
15	10, 14	syl6eq 2672	. . . . . 6 |- ( e = a -> U_ f e. e ( { f } X. ~P f ) = U_ b e. a ( { b } X. ~P b ) )
16	15	fveq2d 6195	. . . . 5 |- ( e = a -> ( card ` U_ f e. e ( { f } X. ~P f ) ) = ( card ` U_ b e. a ( { b } X. ~P b ) ) )
17	16	cbvmptv 4750	. . . 4 |- ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) = ( a e. ( ~P om i^i Fin ) |-> ( card ` U_ b e. a ( { b } X. ~P b ) ) )
18		dmeq 5324	. . . . . . . 8 |- ( c = a -> dom c = dom a )
19	18	pweqd 4163	. . . . . . 7 |- ( c = a -> ~P dom c = ~P dom a )
20		imaeq1 5461	. . . . . . . 8 |- ( c = a -> ( c " d ) = ( a " d ) )
21	20	fveq2d 6195	. . . . . . 7 |- ( c = a -> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( c " d ) ) = ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( a " d ) ) )
22	19, 21	mpteq12dv 4733	. . . . . 6 |- ( c = a -> ( d e. ~P dom c |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( c " d ) ) ) = ( d e. ~P dom a |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( a " d ) ) ) )
23		imaeq2 5462	. . . . . . . 8 |- ( d = b -> ( a " d ) = ( a " b ) )
24	23	fveq2d 6195	. . . . . . 7 |- ( d = b -> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( a " d ) ) = ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( a " b ) ) )
25	24	cbvmptv 4750	. . . . . 6 |- ( d e. ~P dom a |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( a " d ) ) ) = ( b e. ~P dom a |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( a " b ) ) )
26	22, 25	syl6eq 2672	. . . . 5 |- ( c = a -> ( d e. ~P dom c |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( c " d ) ) ) = ( b e. ~P dom a |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( a " b ) ) ) )
27	26	cbvmptv 4750	. . . 4 |- ( c e. _V |-> ( d e. ~P dom c |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( c " d ) ) ) ) = ( a e. _V |-> ( b e. ~P dom a |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( a " b ) ) ) )
28		eqid 2622	. . . 4 |- U. ( rec ( ( c e. _V |-> ( d e. ~P dom c |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( c " d ) ) ) ) , (/) ) " om ) = U. ( rec ( ( c e. _V |-> ( d e. ~P dom c |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( c " d ) ) ) ) , (/) ) " om )
29	17, 27, 28	ackbij2 9065	. . 3 |- U. ( rec ( ( c e. _V |-> ( d e. ~P dom c |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( c " d ) ) ) ) , (/) ) " om ) : U. ( R1 " om ) -1-1-onto-> om
30		fvex 6201	. . . . 5 |- ( R1 ` om ) e. _V
31	9, 30	eqeltrri 2698	. . . 4 |- U. ( R1 " om ) e. _V
32	31	f1oen 7976	. . 3 |- ( U. ( rec ( ( c e. _V |-> ( d e. ~P dom c |-> ( ( e e. ( ~P om i^i Fin ) |-> ( card ` U_ f e. e ( { f } X. ~P f ) ) ) ` ( c " d ) ) ) ) , (/) ) " om ) : U. ( R1 " om ) -1-1-onto-> om -> U. ( R1 " om ) ~~ om )
33	29, 32	ax-mp 5	. 2 |- U. ( R1 " om ) ~~ om
34	9, 33	eqbrtri 4674	1 |- ( R1 ` om ) ~~ om

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   "cima 5117   Oncon0 5723   Lim wlim 5724   Fun wfun 5882   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888   omcom 7065   reccrdg 7505   ~~ cen 7952   Fincfn 7955   R1cr1 8625   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-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-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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-r1 8627  df-rank 8628  df-card 8765  df-cda 8990

This theorem is referenced by: (None)