Theorem rdgss 6232

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 3055	. . . . . 6 |- ( A C_ B -> ( x e. A -> x e. B ) )
3		ssid 3081	. . . . . . 7 |- ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` x ) )
4		fveq2 5373	. . . . . . . . . 10 |- ( y = x -> ( rec ( F , I ) ` y ) = ( rec ( F , I ) ` x ) )
5	4	fveq2d 5377	. . . . . . . . 9 |- ( y = x -> ( F ` ( rec ( F , I ) ` y ) ) = ( F ` ( rec ( F , I ) ` x ) ) )
6	5	sseq2d 3091	. . . . . . . 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 2758	. . . . . . 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 419	. . . . . 6 |- ( x e. B -> E. y e. B ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` y ) ) )
9	2, 8	syl6 33	. . . . 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 2476	. . . 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 3822	. . 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 3211	. . 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 6231	. . 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 1197	. 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 6231	. . 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 1197	. 2 |- ( ph -> ( rec ( F , I ) ` B ) = ( I u. U_ y e. B ( F ` ( rec ( F , I ) ` y ) ) ) )
23	14, 19, 22	3sstr4d 3106	1 |- ( ph -> ( rec ( F , I ) ` A ) C_ ( rec ( F , I ) ` B ) )

Syntax hints:   -> wi 4   = wceq 1312   e. wcel 1461   A.wral 2388   E.wrex 2389   _Vcvv 2655   u. cun 3033   C_ wss 3035   U_ciun 3777   Oncon0 4243   Fn wfn 5074   ` cfv 5079   reccrdg 6218

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-recs 6154  df-irdg 6219

This theorem is referenced by:  oawordi  6317