Theorem fodjumkvlemres 7263

Description: Lemma for fodjumkv 7264. 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 7251	. . . 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 2418	. 2 |- ( ph -> ( A =/= (/) -> -. A. w e. M ( P ` w ) = 1o ) )
8		fveq1 5577	. . . . . . 7 |- ( f = P -> ( f ` w ) = ( P ` w ) )
9	8	eqeq1d 2214	. . . . . 6 |- ( f = P -> ( ( f ` w ) = 1o <-> ( P ` w ) = 1o ) )
10	9	ralbidv 2506	. . . . 5 |- ( f = P -> ( A. w e. M ( f ` w ) = 1o <-> A. w e. M ( P ` w ) = 1o ) )
11	10	notbid 669	. . . 4 |- ( f = P -> ( -. A. w e. M ( f ` w ) = 1o <-> -. A. w e. M ( P ` w ) = 1o ) )
12	8	eqeq1d 2214	. . . . 5 |- ( f = P -> ( ( f ` w ) = (/) <-> ( P ` w ) = (/) ) )
13	12	rexbidv 2507	. . . 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 7258	. . . . 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 7249	. . 3 |- ( ph -> P e. ( 2o ^m M ) )
20	14, 18, 19	rspcdva 2882	. 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 5587	. . . . . 6 |- ( w = v -> ( ( P ` w ) = (/) <-> ( P ` v ) = (/) ) )
24	23	cbvrexv 2739	. . . . 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 7250	. . 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 1373   E.wex 1515   e. wcel 2176   =/= wne 2376   A.wral 2484   E.wrex 2485   (/)c0 3460   ifcif 3571   |-> cmpt 4106   -onto->wfo 5270   ` cfv 5272  (class class class)co 5946   1oc1o 6497   2oc2o 6498   ^m cmap 6737   ⊔ cdju 7141  inlcinl 7149  Markovcmarkov 7255

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-1o 6504  df-2o 6505  df-map 6739  df-dju 7142  df-inl 7151  df-inr 7152  df-markov 7256

This theorem is referenced by:  fodjumkv  7264