Theorem fodjumkvlemres 7499

Description: Lemma for fodjumkv 7500. 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 7487	. . . 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 2462	. 2 |- ( ph -> ( A =/= (/) -> -. A. w e. M ( P ` w ) = 1o ) )
8		fveq1 5694	. . . . . . 7 |- ( f = P -> ( f ` w ) = ( P ` w ) )
9	8	eqeq1d 2247	. . . . . 6 |- ( f = P -> ( ( f ` w ) = 1o <-> ( P ` w ) = 1o ) )
10	9	ralbidv 2550	. . . . 5 |- ( f = P -> ( A. w e. M ( f ` w ) = 1o <-> A. w e. M ( P ` w ) = 1o ) )
11	10	notbid 677	. . . 4 |- ( f = P -> ( -. A. w e. M ( f ` w ) = 1o <-> -. A. w e. M ( P ` w ) = 1o ) )
12	8	eqeq1d 2247	. . . . 5 |- ( f = P -> ( ( f ` w ) = (/) <-> ( P ` w ) = (/) ) )
13	12	rexbidv 2551	. . . 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 7494	. . . . 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 7485	. . 3 |- ( ph -> P e. ( 2o ^m M ) )
20	14, 18, 19	rspcdva 2934	. 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 5704	. . . . . 6 |- ( w = v -> ( ( P ` w ) = (/) <-> ( P ` v ) = (/) ) )
24	23	cbvrexv 2787	. . . . 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 7486	. . 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 ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   A.wral 2528   E.wrex 2529   (/)c0 3520   ifcif 3638   |-> cmpt 4192   -onto->wfo 5375   ` cfv 5377  (class class class)co 6085   1oc1o 6680   2oc2o 6681   ^m cmap 6922   ⊔ cdju 7377  inlcinl 7385  Markovcmarkov 7491

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-map 6924  df-dju 7378  df-inl 7387  df-inr 7388  df-markov 7492

This theorem is used by:  fodjumkv  7500