Theorem ov2ssiunov2 37992

Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 13798 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)

Hypothesis

Ref	Expression
ov2ssiunov2.def	|- C = ( r e. _V |-> U_ n e. N ( r .^ n ) )

Assertion

Ref	Expression
ov2ssiunov2	|- ( ( R e. U /\ N e. V /\ M e. N ) -> ( R .^ M ) C_ ( C ` R ) )

Distinct variable groups:   n, r, C, N, .^   n, M   R, r, n   U, n   n, V

Allowed substitution hints:   U( r)   M( r)   V( r)

Proof of Theorem ov2ssiunov2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1063	. . . 4 |- ( ( R e. U /\ N e. V /\ M e. N ) -> M e. N )
2		simpr 477	. . . . . 6 |- ( ( ( R e. U /\ N e. V /\ M e. N ) /\ n = M ) -> n = M )
3	2	oveq2d 6666	. . . . 5 |- ( ( ( R e. U /\ N e. V /\ M e. N ) /\ n = M ) -> ( R .^ n ) = ( R .^ M ) )
4	3	eleq2d 2687	. . . 4 |- ( ( ( R e. U /\ N e. V /\ M e. N ) /\ n = M ) -> ( x e. ( R .^ n ) <-> x e. ( R .^ M ) ) )
5	1, 4	rspcedv 3313	. . 3 |- ( ( R e. U /\ N e. V /\ M e. N ) -> ( x e. ( R .^ M ) -> E. n e. N x e. ( R .^ n ) ) )
6		ov2ssiunov2.def	. . . . . 6 |- C = ( r e. _V |-> U_ n e. N ( r .^ n ) )
7	6	eliunov2 37971	. . . . 5 |- ( ( R e. U /\ N e. V ) -> ( x e. ( C ` R ) <-> E. n e. N x e. ( R .^ n ) ) )
8	7	biimprd 238	. . . 4 |- ( ( R e. U /\ N e. V ) -> ( E. n e. N x e. ( R .^ n ) -> x e. ( C ` R ) ) )
9	8	3adant3 1081	. . 3 |- ( ( R e. U /\ N e. V /\ M e. N ) -> ( E. n e. N x e. ( R .^ n ) -> x e. ( C ` R ) ) )
10	5, 9	syld 47	. 2 |- ( ( R e. U /\ N e. V /\ M e. N ) -> ( x e. ( R .^ M ) -> x e. ( C ` R ) ) )
11	10	ssrdv 3609	1 |- ( ( R e. U /\ N e. V /\ M e. N ) -> ( R .^ M ) C_ ( C ` R ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   U_ciun 4520   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650

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-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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

This theorem is referenced by:  dftrcl3  38012  dfrtrcl3  38025