Theorem fodjumkvlemres 7150

Description: Lemma for fodjumkv 7151. The final result with P expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)

Hypotheses

Ref	Expression
fodjumkv.o	|- ( ph -> M e. Markov )
fodjumkv.fo	|- ( ph -> F : M -onto-> ( A ⊔ B ) )
fodjumkv.p	|- P = ( y e. M |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )

Assertion

Ref	Expression
fodjumkvlemres	|- ( ph -> ( A =/= (/) -> E. x x e. A ) )

Distinct variable groups:   ph, y, z   y, M, z   z, A   z, B   z, F   x, A, z   y, A   y, F   y, P, z

Allowed substitution hints:   ph( x)   B( x, y)   P( x)   F( x)   M( x)

Proof of Theorem fodjumkvlemres

Dummy variables v f w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjumkv.fo	. . . . . 6 |- ( ph -> F : M -onto-> ( A ⊔ B ) )
2	1	adantr 276	. . . . 5 |- ( ( ph /\ A. w e. M ( P ` w ) = 1o ) -> F : M -onto-> ( A ⊔ B ) )
3		fodjumkv.p	. . . . 5 |- P = ( y e. M |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
4		simpr 110	. . . . 5 |- ( ( ph /\ A. w e. M ( P ` w ) = 1o ) -> A. w e. M ( P ` w ) = 1o )
5	2, 3, 4	fodju0 7138	. . . 4 |- ( ( ph /\ A. w e. M ( P ` w ) = 1o ) -> A = (/) )
6	5	ex 115	. . 3 |- ( ph -> ( A. w e. M ( P ` w ) = 1o -> A = (/) ) )
7	6	necon3ad 2389	. 2 |- ( ph -> ( A =/= (/) -> -. A. w e. M ( P ` w ) = 1o ) )
8		fveq1 5509	. . . . . . 7 |- ( f = P -> ( f ` w ) = ( P ` w ) )
9	8	eqeq1d 2186	. . . . . 6 |- ( f = P -> ( ( f ` w ) = 1o <-> ( P ` w ) = 1o ) )
10	9	ralbidv 2477	. . . . 5 |- ( f = P -> ( A. w e. M ( f ` w ) = 1o <-> A. w e. M ( P ` w ) = 1o ) )
11	10	notbid 667	. . . 4 |- ( f = P -> ( -. A. w e. M ( f ` w ) = 1o <-> -. A. w e. M ( P ` w ) = 1o ) )
12	8	eqeq1d 2186	. . . . 5 |- ( f = P -> ( ( f ` w ) = (/) <-> ( P ` w ) = (/) ) )
13	12	rexbidv 2478	. . . 4 |- ( f = P -> ( E. w e. M ( f ` w ) = (/) <-> E. w e. M ( P ` w ) = (/) ) )
14	11, 13	imbi12d 234	. . 3 |- ( f = P -> ( ( -. A. w e. M ( f ` w ) = 1o -> E. w e. M ( f ` w ) = (/) ) <-> ( -. A. w e. M ( P ` w ) = 1o -> E. w e. M ( P ` w ) = (/) ) ) )
15		fodjumkv.o	. . . 4 |- ( ph -> M e. Markov )
16		ismkvmap 7145	. . . . 5 |- ( M e. Markov -> ( M e. Markov <-> A. f e. ( 2o ^m M ) ( -. A. w e. M ( f ` w ) = 1o -> E. w e. M ( f ` w ) = (/) ) ) )
17	16	ibi 176	. . . 4 |- ( M e. Markov -> A. f e. ( 2o ^m M ) ( -. A. w e. M ( f ` w ) = 1o -> E. w e. M ( f ` w ) = (/) ) )
18	15, 17	syl 14	. . 3 |- ( ph -> A. f e. ( 2o ^m M ) ( -. A. w e. M ( f ` w ) = 1o -> E. w e. M ( f ` w ) = (/) ) )
19	1, 3, 15	fodjuf 7136	. . 3 |- ( ph -> P e. ( 2o ^m M ) )
20	14, 18, 19	rspcdva 2846	. 2 |- ( ph -> ( -. A. w e. M ( P ` w ) = 1o -> E. w e. M ( P ` w ) = (/) ) )
21	1	adantr 276	. . . 4 |- ( ( ph /\ E. w e. M ( P ` w ) = (/) ) -> F : M -onto-> ( A ⊔ B ) )
22		simpr 110	. . . . 5 |- ( ( ph /\ E. w e. M ( P ` w ) = (/) ) -> E. w e. M ( P ` w ) = (/) )
23		fveqeq2 5519	. . . . . 6 |- ( w = v -> ( ( P ` w ) = (/) <-> ( P ` v ) = (/) ) )
24	23	cbvrexv 2704	. . . . 5 |- ( E. w e. M ( P ` w ) = (/) <-> E. v e. M ( P ` v ) = (/) )
25	22, 24	sylib 122	. . . 4 |- ( ( ph /\ E. w e. M ( P ` w ) = (/) ) -> E. v e. M ( P ` v ) = (/) )
26	21, 3, 25	fodjum 7137	. . 3 |- ( ( ph /\ E. w e. M ( P ` w ) = (/) ) -> E. x x e. A )
27	26	ex 115	. 2 |- ( ph -> ( E. w e. M ( P ` w ) = (/) -> E. x x e. A ) )
28	7, 20, 27	3syld 57	1 |- ( ph -> ( A =/= (/) -> E. x x e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   =/= wne 2347   A.wral 2455   E.wrex 2456   (/)c0 3422   ifcif 3534   |-> cmpt 4061   -onto->wfo 5209   ` cfv 5211  (class class class)co 5868   1oc1o 6403   2oc2o 6404   ^m cmap 6641   ⊔ cdju 7029  inlcinl 7037  Markovcmarkov 7142

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-1o 6410  df-2o 6411  df-map 6643  df-dju 7030  df-inl 7039  df-inr 7040  df-markov 7143

This theorem is referenced by:  fodjumkv  7151