Theorem relexpss1d 36343

Description: The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)

Hypotheses

Ref	Expression
relexpss1d.a	|- ( ph -> A C_ B )
relexpss1d.b	|- ( ph -> B e. _V )
relexpss1d.n	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
relexpss1d	|- ( ph -> ( A ^r N ) C_ ( B ^r N ) )

Proof of Theorem relexpss1d

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

Step	Hyp	Ref	Expression
1		relexpss1d.n	. . 3 |- ( ph -> N e. NN0 )
2		elnn0 10905	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
3	1, 2	sylib 201	. 2 |- ( ph -> ( N e. NN \/ N = 0 ) )
4		oveq2 6328	. . . . . 6 |- ( x = 1 -> ( A ^r x ) = ( A ^r 1 ) )
5		oveq2 6328	. . . . . 6 |- ( x = 1 -> ( B ^r x ) = ( B ^r 1 ) )
6	4, 5	sseq12d 3473	. . . . 5 |- ( x = 1 -> ( ( A ^r x ) C_ ( B ^r x ) <-> ( A ^r 1 ) C_ ( B ^r 1 ) ) )
7	6	imbi2d 322	. . . 4 |- ( x = 1 -> ( ( ph -> ( A ^r x ) C_ ( B ^r x ) ) <-> ( ph -> ( A ^r 1 ) C_ ( B ^r 1 ) ) ) )
8		oveq2 6328	. . . . . 6 |- ( x = y -> ( A ^r x ) = ( A ^r y ) )
9		oveq2 6328	. . . . . 6 |- ( x = y -> ( B ^r x ) = ( B ^r y ) )
10	8, 9	sseq12d 3473	. . . . 5 |- ( x = y -> ( ( A ^r x ) C_ ( B ^r x ) <-> ( A ^r y ) C_ ( B ^r y ) ) )
11	10	imbi2d 322	. . . 4 |- ( x = y -> ( ( ph -> ( A ^r x ) C_ ( B ^r x ) ) <-> ( ph -> ( A ^r y ) C_ ( B ^r y ) ) ) )
12		oveq2 6328	. . . . . 6 |- ( x = ( y + 1 ) -> ( A ^r x ) = ( A ^r ( y + 1 ) ) )
13		oveq2 6328	. . . . . 6 |- ( x = ( y + 1 ) -> ( B ^r x ) = ( B ^r ( y + 1 ) ) )
14	12, 13	sseq12d 3473	. . . . 5 |- ( x = ( y + 1 ) -> ( ( A ^r x ) C_ ( B ^r x ) <-> ( A ^r ( y + 1 ) ) C_ ( B ^r ( y + 1 ) ) ) )
15	14	imbi2d 322	. . . 4 |- ( x = ( y + 1 ) -> ( ( ph -> ( A ^r x ) C_ ( B ^r x ) ) <-> ( ph -> ( A ^r ( y + 1 ) ) C_ ( B ^r ( y + 1 ) ) ) ) )
16		oveq2 6328	. . . . . 6 |- ( x = N -> ( A ^r x ) = ( A ^r N ) )
17		oveq2 6328	. . . . . 6 |- ( x = N -> ( B ^r x ) = ( B ^r N ) )
18	16, 17	sseq12d 3473	. . . . 5 |- ( x = N -> ( ( A ^r x ) C_ ( B ^r x ) <-> ( A ^r N ) C_ ( B ^r N ) ) )
19	18	imbi2d 322	. . . 4 |- ( x = N -> ( ( ph -> ( A ^r x ) C_ ( B ^r x ) ) <-> ( ph -> ( A ^r N ) C_ ( B ^r N ) ) ) )
20		relexpss1d.a	. . . . 5 |- ( ph -> A C_ B )
21		relexpss1d.b	. . . . . . 7 |- ( ph -> B e. _V )
22	21, 20	ssexd 4566	. . . . . 6 |- ( ph -> A e. _V )
23	22	relexp1d 13149	. . . . 5 |- ( ph -> ( A ^r 1 ) = A )
24	21	relexp1d 13149	. . . . 5 |- ( ph -> ( B ^r 1 ) = B )
25	20, 23, 24	3sstr4d 3487	. . . 4 |- ( ph -> ( A ^r 1 ) C_ ( B ^r 1 ) )
26		simp3 1016	. . . . . . . 8 |- ( ( y e. NN /\ ph /\ ( A ^r y ) C_ ( B ^r y ) ) -> ( A ^r y ) C_ ( B ^r y ) )
27	20	3ad2ant2 1036	. . . . . . . 8 |- ( ( y e. NN /\ ph /\ ( A ^r y ) C_ ( B ^r y ) ) -> A C_ B )
28	26, 27	coss12d 13091	. . . . . . 7 |- ( ( y e. NN /\ ph /\ ( A ^r y ) C_ ( B ^r y ) ) -> ( ( A ^r y ) o. A ) C_ ( ( B ^r y ) o. B ) )
29	22	3ad2ant2 1036	. . . . . . . 8 |- ( ( y e. NN /\ ph /\ ( A ^r y ) C_ ( B ^r y ) ) -> A e. _V )
30		simp1 1014	. . . . . . . 8 |- ( ( y e. NN /\ ph /\ ( A ^r y ) C_ ( B ^r y ) ) -> y e. NN )
31		relexpsucnnr 13143	. . . . . . . 8 |- ( ( A e. _V /\ y e. NN ) -> ( A ^r ( y + 1 ) ) = ( ( A ^r y ) o. A ) )
32	29, 30, 31	syl2anc 671	. . . . . . 7 |- ( ( y e. NN /\ ph /\ ( A ^r y ) C_ ( B ^r y ) ) -> ( A ^r ( y + 1 ) ) = ( ( A ^r y ) o. A ) )
33	21	3ad2ant2 1036	. . . . . . . 8 |- ( ( y e. NN /\ ph /\ ( A ^r y ) C_ ( B ^r y ) ) -> B e. _V )
34		relexpsucnnr 13143	. . . . . . . 8 |- ( ( B e. _V /\ y e. NN ) -> ( B ^r ( y + 1 ) ) = ( ( B ^r y ) o. B ) )
35	33, 30, 34	syl2anc 671	. . . . . . 7 |- ( ( y e. NN /\ ph /\ ( A ^r y ) C_ ( B ^r y ) ) -> ( B ^r ( y + 1 ) ) = ( ( B ^r y ) o. B ) )
36	28, 32, 35	3sstr4d 3487	. . . . . 6 |- ( ( y e. NN /\ ph /\ ( A ^r y ) C_ ( B ^r y ) ) -> ( A ^r ( y + 1 ) ) C_ ( B ^r ( y + 1 ) ) )
37	36	3exp 1214	. . . . 5 |- ( y e. NN -> ( ph -> ( ( A ^r y ) C_ ( B ^r y ) -> ( A ^r ( y + 1 ) ) C_ ( B ^r ( y + 1 ) ) ) ) )
38	37	a2d 29	. . . 4 |- ( y e. NN -> ( ( ph -> ( A ^r y ) C_ ( B ^r y ) ) -> ( ph -> ( A ^r ( y + 1 ) ) C_ ( B ^r ( y + 1 ) ) ) ) )
39	7, 11, 15, 19, 25, 38	nnind 10660	. . 3 |- ( N e. NN -> ( ph -> ( A ^r N ) C_ ( B ^r N ) ) )
40		simpr 467	. . . . . 6 |- ( ( N = 0 /\ ph ) -> ph )
41		dmss 5056	. . . . . . . 8 |- ( A C_ B -> dom A C_ dom B )
42		rnss 5085	. . . . . . . 8 |- ( A C_ B -> ran A C_ ran B )
43	41, 42	jca 539	. . . . . . 7 |- ( A C_ B -> ( dom A C_ dom B /\ ran A C_ ran B ) )
44		unss12 3618	. . . . . . 7 |- ( ( dom A C_ dom B /\ ran A C_ ran B ) -> ( dom A u. ran A ) C_ ( dom B u. ran B ) )
45	20, 43, 44	3syl 18	. . . . . 6 |- ( ph -> ( dom A u. ran A ) C_ ( dom B u. ran B ) )
46		ssres2 5153	. . . . . 6 |- ( ( dom A u. ran A ) C_ ( dom B u. ran B ) -> ( _I |` ( dom A u. ran A ) ) C_ ( _I |` ( dom B u. ran B ) ) )
47	40, 45, 46	3syl 18	. . . . 5 |- ( ( N = 0 /\ ph ) -> ( _I |` ( dom A u. ran A ) ) C_ ( _I |` ( dom B u. ran B ) ) )
48		simpl 463	. . . . . . 7 |- ( ( N = 0 /\ ph ) -> N = 0 )
49	48	oveq2d 6336	. . . . . 6 |- ( ( N = 0 /\ ph ) -> ( A ^r N ) = ( A ^r 0 ) )
50		relexp0g 13140	. . . . . . 7 |- ( A e. _V -> ( A ^r 0 ) = ( _I |` ( dom A u. ran A ) ) )
51	40, 22, 50	3syl 18	. . . . . 6 |- ( ( N = 0 /\ ph ) -> ( A ^r 0 ) = ( _I |` ( dom A u. ran A ) ) )
52	49, 51	eqtrd 2496	. . . . 5 |- ( ( N = 0 /\ ph ) -> ( A ^r N ) = ( _I |` ( dom A u. ran A ) ) )
53	48	oveq2d 6336	. . . . . 6 |- ( ( N = 0 /\ ph ) -> ( B ^r N ) = ( B ^r 0 ) )
54		relexp0g 13140	. . . . . . 7 |- ( B e. _V -> ( B ^r 0 ) = ( _I |` ( dom B u. ran B ) ) )
55	40, 21, 54	3syl 18	. . . . . 6 |- ( ( N = 0 /\ ph ) -> ( B ^r 0 ) = ( _I |` ( dom B u. ran B ) ) )
56	53, 55	eqtrd 2496	. . . . 5 |- ( ( N = 0 /\ ph ) -> ( B ^r N ) = ( _I |` ( dom B u. ran B ) ) )
57	47, 52, 56	3sstr4d 3487	. . . 4 |- ( ( N = 0 /\ ph ) -> ( A ^r N ) C_ ( B ^r N ) )
58	57	ex 440	. . 3 |- ( N = 0 -> ( ph -> ( A ^r N ) C_ ( B ^r N ) ) )
59	39, 58	jaoi 385	. 2 |- ( ( N e. NN \/ N = 0 ) -> ( ph -> ( A ^r N ) C_ ( B ^r N ) ) )
60	3, 59	mpcom 37	1 |- ( ph -> ( A ^r N ) C_ ( B ^r N ) )

Syntax hints:   -> wi 4   \/ wo 374   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   _Vcvv 3057   u. cun 3414   C_ wss 3416   _I cid 4766   dom cdm 4856   ran crn 4857   |` cres 4858   o. ccom 4860  (class class class)co 6320   0cc0 9570   1c1 9571   + caddc 9573   NNcn 10642   NN0cn0 10903   ^r crelexp 13138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-n0 10904  df-z 10972  df-uz 11194  df-seq 12252  df-relexp 13139

This theorem is referenced by:  corcltrcl  36377  cotrclrcl  36380