Theorem rdgss 6450

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 3178	. . . . . 6 |- ( A C_ B -> ( x e. A -> x e. B ) )
3		ssid 3204	. . . . . . 7 |- ( F ` ( rec ( F , I ) ` x ) ) C_ ( F ` ( rec ( F , I ) ` x ) )
4		fveq2 5561	. . . . . . . . . 10 |- ( y = x -> ( rec ( F , I ) ` y ) = ( rec ( F , I ) ` x ) )
5	4	fveq2d 5565	. . . . . . . . 9 |- ( y = x -> ( F ` ( rec ( F , I ) ` y ) ) = ( F ` ( rec ( F , I ) ` x ) ) )
6	5	sseq2d 3214	. . . . . . . 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 2868	. . . . . . 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 425	. . . . . 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 2569	. . . 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 3962	. . 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 3335	. . 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 6449	. . 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 1249	. 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 6449	. . 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 1249	. 2 |- ( ph -> ( rec ( F , I ) ` B ) = ( I u. U_ y e. B ( F ` ( rec ( F , I ) ` y ) ) ) )
23	14, 19, 22	3sstr4d 3229	1 |- ( ph -> ( rec ( F , I ) ` A ) C_ ( rec ( F , I ) ` B ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   _Vcvv 2763   u. cun 3155   C_ wss 3157   U_ciun 3917   Oncon0 4399   Fn wfn 5254   ` cfv 5259   reccrdg 6436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-recs 6372  df-irdg 6437

This theorem is referenced by:  oawordi  6536