Theorem rdgss 5970

Description: Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)

Hypotheses

Ref	Expression
rdgss.1	|- ( ph -> F Fn _V )
rdgss.2	|- ( ph -> I e. V )
rdgss.3	|- ( ph -> A e. On )
rdgss.4	|- ( ph -> B e. On )
rdgss.5	|- ( ph -> A C_ B )

Assertion

Ref	Expression
rdgss	|- ( ph -> ( rec ( F , I ) ` A ) C_ ( rec ( F , I ) ` B ) )

Proof of Theorem rdgss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdgss.5	. . . 4 |- ( ph -> A C_ B )
2		ssel 2939	. . . . . 6 |- ( A C_ B -> ( x e. A -> x e. B ) )
3		ssid 2964	. . . . . . 7 |- ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` x ) )
4		fveq2 5178	. . . . . . . . . 10 |- ( y = x -> ( rec ( F , I ) ` y ) = ( rec ( F , I ) ` x ) )
5	4	fveq2d 5182	. . . . . . . . 9 |- ( y = x -> ( F ` ( rec ( F , I ) ` y ) ) = ( F ` ( rec ( F , I ) ` x ) ) )
6	5	sseq2d 2973	. . . . . . . 8 |- ( y = x -> ( ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` y ) ) <-> ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` x ) ) ) )
7	6	rspcev 2656	. . . . . . 7 |- ( ( x e. B /\ ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` x ) ) ) -> E. y e. B ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` y ) ) )
8	3, 7	mpan2 401	. . . . . 6 |- ( x e. B -> E. y e. B ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` y ) ) )
9	2, 8	syl6 29	. . . . 5 |- ( A C_ B -> ( x e. A -> E. y e. B ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` y ) ) ) )
10	9	ralrimiv 2391	. . . 4 |- ( A C_ B -> A. x e. A E. y e. B ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` y ) ) )
11	1, 10	syl 14	. . 3 |- ( ph -> A. x e. A E. y e. B ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` y ) ) )
12		iunss2 3702	. . 3 |- ( A. x e. A E. y e. B ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` y ) ) -> U_ x e. A ( F ` ( rec ( F , I ) ` x ) ) C_ U_ y e. B ( F ` ( rec ( F , I ) ` y ) ) )
13		unss2 3114	. . 3 |- ( U_ x e. A ( F ` ( rec ( F , I ) ` x ) ) C_ U_ y e. B ( F ` ( rec ( F , I ) ` y ) ) -> ( I u. U_ x e. A ( F ` ( rec ( F , I ) ` x ) ) ) C_ ( I u. U_ y e. B ( F ` ( rec ( F , I ) ` y ) ) ) )
14	11, 12, 13	3syl 17	. 2 |- ( ph -> ( I u. U_ x e. A ( F ` ( rec ( F , I ) ` x ) ) ) C_ ( I u. U_ y e. B ( F ` ( rec ( F , I ) ` y ) ) ) )
15		rdgss.1	. . 3 |- ( ph -> F Fn _V )
16		rdgss.2	. . 3 |- ( ph -> I e. V )
17		rdgss.3	. . 3 |- ( ph -> A e. On )
18		rdgival 5969	. . 3 |- ( ( F Fn _V /\ I e. V /\ A e. On ) -> ( rec ( F , I ) ` A ) = ( I u. U_ x e. A ( F ` ( rec ( F , I ) ` x ) ) ) )
19	15, 16, 17, 18	syl3anc 1135	. 2 |- ( ph -> ( rec ( F , I ) ` A ) = ( I u. U_ x e. A ( F ` ( rec ( F , I ) ` x ) ) ) )
20		rdgss.4	. . 3 |- ( ph -> B e. On )
21		rdgival 5969	. . 3 |- ( ( F Fn _V /\ I e. V /\ B e. On ) -> ( rec ( F , I ) ` B ) = ( I u. U_ y e. B ( F ` ( rec ( F , I ) ` y ) ) ) )
22	15, 16, 20, 21	syl3anc 1135	. 2 |- ( ph -> ( rec ( F , I ) ` B ) = ( I u. U_ y e. B ( F ` ( rec ( F , I ) ` y ) ) ) )
23	14, 19, 22	3sstr4d 2988	1 |- ( ph -> ( rec ( F , I ) ` A ) C_ ( rec ( F , I ) ` B ) )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   A.wral 2306   E.wrex 2307   _Vcvv 2557   u. cun 2915   C_ wss 2917   U_ciun 3657   Oncon0 4100   Fn wfn 4897   ` cfv 4902   reccrdg 5956

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-irdg 5957

This theorem is referenced by:  oawordi  6049