Theorem fodjumkvlemres 7234

Description: Lemma for fodjumkv 7235. 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 7222	. . . 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 2409	. 2 |- ( ph -> ( A =/= (/) -> -. A. w e. M ( P ` w ) = 1o ) )
8		fveq1 5560	. . . . . . 7 |- ( f = P -> ( f ` w ) = ( P ` w ) )
9	8	eqeq1d 2205	. . . . . 6 |- ( f = P -> ( ( f ` w ) = 1o <-> ( P ` w ) = 1o ) )
10	9	ralbidv 2497	. . . . 5 |- ( f = P -> ( A. w e. M ( f ` w ) = 1o <-> A. w e. M ( P ` w ) = 1o ) )
11	10	notbid 668	. . . 4 |- ( f = P -> ( -. A. w e. M ( f ` w ) = 1o <-> -. A. w e. M ( P ` w ) = 1o ) )
12	8	eqeq1d 2205	. . . . 5 |- ( f = P -> ( ( f ` w ) = (/) <-> ( P ` w ) = (/) ) )
13	12	rexbidv 2498	. . . 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 7229	. . . . 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 7220	. . 3 |- ( ph -> P e. ( 2o ^m M ) )
20	14, 18, 19	rspcdva 2873	. 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 5570	. . . . . 6 |- ( w = v -> ( ( P ` w ) = (/) <-> ( P ` v ) = (/) ) )
24	23	cbvrexv 2730	. . . . 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 7221	. . 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 1364   E.wex 1506   e. wcel 2167   =/= wne 2367   A.wral 2475   E.wrex 2476   (/)c0 3451   ifcif 3562   |-> cmpt 4095   -onto->wfo 5257   ` cfv 5259  (class class class)co 5925   1oc1o 6476   2oc2o 6477   ^m cmap 6716   ⊔ cdju 7112  inlcinl 7120  Markovcmarkov 7226

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-1o 6483  df-2o 6484  df-map 6718  df-dju 7113  df-inl 7122  df-inr 7123  df-markov 7227

This theorem is referenced by:  fodjumkv  7235