Theorem fvmptiunrelexplb0d 37976

Description: If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)

Hypotheses

Ref	Expression
fvmptiunrelexplb0d.c	|- C = ( r e. _V |-> U_ n e. N ( r ^r n ) )
fvmptiunrelexplb0d.r	|- ( ph -> R e. _V )
fvmptiunrelexplb0d.n	|- ( ph -> N e. _V )
fvmptiunrelexplb0d.0	|- ( ph -> 0 e. N )

Assertion

Ref	Expression
fvmptiunrelexplb0d	|- ( ph -> ( _I |` ( dom R u. ran R ) ) C_ ( C ` R ) )

Distinct variable groups:   n, r, N   R, n, r

Allowed substitution hints:   ph( n, r)   C( n, r)

Proof of Theorem fvmptiunrelexplb0d

Step	Hyp	Ref	Expression
1		fvmptiunrelexplb0d.0	. . 3 |- ( ph -> 0 e. N )
2		oveq2 6658	. . . 4 |- ( n = 0 -> ( R ^r n ) = ( R ^r 0 ) )
3	2	ssiun2s 4564	. . 3 |- ( 0 e. N -> ( R ^r 0 ) C_ U_ n e. N ( R ^r n ) )
4	1, 3	syl 17	. 2 |- ( ph -> ( R ^r 0 ) C_ U_ n e. N ( R ^r n ) )
5		fvmptiunrelexplb0d.r	. . 3 |- ( ph -> R e. _V )
6		relexp0g 13762	. . 3 |- ( R e. _V -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) )
7	5, 6	syl 17	. 2 |- ( ph -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) )
8		fvmptiunrelexplb0d.n	. . . . 5 |- ( ph -> N e. _V )
9		ovex 6678	. . . . . 6 |- ( R ^r n ) e. _V
10	9	rgenw 2924	. . . . 5 |- A. n e. N ( R ^r n ) e. _V
11		iunexg 7143	. . . . 5 |- ( ( N e. _V /\ A. n e. N ( R ^r n ) e. _V ) -> U_ n e. N ( R ^r n ) e. _V )
12	8, 10, 11	sylancl 694	. . . 4 |- ( ph -> U_ n e. N ( R ^r n ) e. _V )
13		oveq1 6657	. . . . . 6 |- ( r = R -> ( r ^r n ) = ( R ^r n ) )
14	13	iuneq2d 4547	. . . . 5 |- ( r = R -> U_ n e. N ( r ^r n ) = U_ n e. N ( R ^r n ) )
15		fvmptiunrelexplb0d.c	. . . . 5 |- C = ( r e. _V |-> U_ n e. N ( r ^r n ) )
16	14, 15	fvmptg 6280	. . . 4 |- ( ( R e. _V /\ U_ n e. N ( R ^r n ) e. _V ) -> ( C ` R ) = U_ n e. N ( R ^r n ) )
17	5, 12, 16	syl2anc 693	. . 3 |- ( ph -> ( C ` R ) = U_ n e. N ( R ^r n ) )
18	17	eqcomd 2628	. 2 |- ( ph -> U_ n e. N ( R ^r n ) = ( C ` R ) )
19	4, 7, 18	3sstr3d 3647	1 |- ( ph -> ( _I |` ( dom R u. ran R ) ) C_ ( C ` R ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   C_ wss 3574   U_ciun 4520   |-> cmpt 4729   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650   0cc0 9936   ^r crelexp 13760

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-pow 4843  ax-pr 4906  ax-un 6949  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

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-pw 4160  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  df-oprab 6654  df-mpt2 6655  df-n0 11293  df-relexp 13761

This theorem is referenced by:  fvmptiunrelexplb0da  37977  fvrcllb0d  37985  fvrtrcllb0d  38027