Theorem 0pledm 23440

Description: Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
0pledm.1	|- ( ph -> A C_ CC )
0pledm.2	|- ( ph -> F Fn A )

Assertion

Ref	Expression
0pledm	|- ( ph -> ( 0p oR <_ F <-> ( A X. { 0 } ) oR <_ F ) )

Proof of Theorem 0pledm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0pledm.1	. . . 4 |- ( ph -> A C_ CC )
2		sseqin2 3817	. . . 4 |- ( A C_ CC <-> ( CC i^i A ) = A )
3	1, 2	sylib 208	. . 3 |- ( ph -> ( CC i^i A ) = A )
4	3	raleqdv 3144	. 2 |- ( ph -> ( A. x e. ( CC i^i A ) 0 <_ ( F ` x ) <-> A. x e. A 0 <_ ( F ` x ) ) )
5		0cn 10032	. . . . . 6 |- 0 e. CC
6		fnconstg 6093	. . . . . 6 |- ( 0 e. CC -> ( CC X. { 0 } ) Fn CC )
7	5, 6	ax-mp 5	. . . . 5 |- ( CC X. { 0 } ) Fn CC
8		df-0p 23437	. . . . . 6 |- 0p = ( CC X. { 0 } )
9	8	fneq1i 5985	. . . . 5 |- ( 0p Fn CC <-> ( CC X. { 0 } ) Fn CC )
10	7, 9	mpbir 221	. . . 4 |- 0p Fn CC
11	10	a1i 11	. . 3 |- ( ph -> 0p Fn CC )
12		0pledm.2	. . 3 |- ( ph -> F Fn A )
13		cnex 10017	. . . 4 |- CC e. _V
14	13	a1i 11	. . 3 |- ( ph -> CC e. _V )
15		ssexg 4804	. . . 4 |- ( ( A C_ CC /\ CC e. _V ) -> A e. _V )
16	1, 13, 15	sylancl 694	. . 3 |- ( ph -> A e. _V )
17		eqid 2622	. . 3 |- ( CC i^i A ) = ( CC i^i A )
18		0pval 23438	. . . 4 |- ( x e. CC -> ( 0p ` x ) = 0 )
19	18	adantl 482	. . 3 |- ( ( ph /\ x e. CC ) -> ( 0p ` x ) = 0 )
20		eqidd 2623	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
21	11, 12, 14, 16, 17, 19, 20	ofrfval 6905	. 2 |- ( ph -> ( 0p oR <_ F <-> A. x e. ( CC i^i A ) 0 <_ ( F ` x ) ) )
22		fnconstg 6093	. . . . 5 |- ( 0 e. CC -> ( A X. { 0 } ) Fn A )
23	5, 22	ax-mp 5	. . . 4 |- ( A X. { 0 } ) Fn A
24	23	a1i 11	. . 3 |- ( ph -> ( A X. { 0 } ) Fn A )
25		inidm 3822	. . 3 |- ( A i^i A ) = A
26		c0ex 10034	. . . . 5 |- 0 e. _V
27	26	fvconst2 6469	. . . 4 |- ( x e. A -> ( ( A X. { 0 } ) ` x ) = 0 )
28	27	adantl 482	. . 3 |- ( ( ph /\ x e. A ) -> ( ( A X. { 0 } ) ` x ) = 0 )
29	24, 12, 16, 16, 25, 28, 20	ofrfval 6905	. 2 |- ( ph -> ( ( A X. { 0 } ) oR <_ F <-> A. x e. A 0 <_ ( F ` x ) ) )
30	4, 21, 29	3bitr4d 300	1 |- ( ph -> ( 0p oR <_ F <-> ( A X. { 0 } ) oR <_ F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   ` cfv 5888   oRcofr 6896   CCcc 9934   0cc0 9936   <_ cle 10075   0pc0p 23436

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-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-cnex 9992  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-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-ofr 6898  df-0p 23437

This theorem is referenced by:  xrge0f  23498  itg20  23504  itg2const  23507  i1fibl  23574  itgitg1  23575  ftc1anclem5  33489  ftc1anclem7  33491