Theorem fodjuomnilemres 7207

Description: Lemma for fodjuomni 7208. The final result with P expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.)

Hypotheses

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

Assertion

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

Distinct variable groups:   ph, y, z   y, O, 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)   O( x)

Proof of Theorem fodjuomnilemres

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

Step	Hyp	Ref	Expression
1		fveq1 5553	. . . . . 6 |- ( f = P -> ( f ` w ) = ( P ` w ) )
2	1	eqeq1d 2202	. . . . 5 |- ( f = P -> ( ( f ` w ) = (/) <-> ( P ` w ) = (/) ) )
3	2	rexbidv 2495	. . . 4 |- ( f = P -> ( E. w e. O ( f ` w ) = (/) <-> E. w e. O ( P ` w ) = (/) ) )
4	1	eqeq1d 2202	. . . . 5 |- ( f = P -> ( ( f ` w ) = 1o <-> ( P ` w ) = 1o ) )
5	4	ralbidv 2494	. . . 4 |- ( f = P -> ( A. w e. O ( f ` w ) = 1o <-> A. w e. O ( P ` w ) = 1o ) )
6	3, 5	orbi12d 794	. . 3 |- ( f = P -> ( ( E. w e. O ( f ` w ) = (/) \/ A. w e. O ( f ` w ) = 1o ) <-> ( E. w e. O ( P ` w ) = (/) \/ A. w e. O ( P ` w ) = 1o ) ) )
7		fodjuomni.o	. . . 4 |- ( ph -> O e. Omni )
8		isomnimap 7196	. . . . 5 |- ( O e. Omni -> ( O e. Omni <-> A. f e. ( 2o ^m O ) ( E. w e. O ( f ` w ) = (/) \/ A. w e. O ( f ` w ) = 1o ) ) )
9	7, 8	syl 14	. . . 4 |- ( ph -> ( O e. Omni <-> A. f e. ( 2o ^m O ) ( E. w e. O ( f ` w ) = (/) \/ A. w e. O ( f ` w ) = 1o ) ) )
10	7, 9	mpbid 147	. . 3 |- ( ph -> A. f e. ( 2o ^m O ) ( E. w e. O ( f ` w ) = (/) \/ A. w e. O ( f ` w ) = 1o ) )
11		fodjuomni.fo	. . . 4 |- ( ph -> F : O -onto-> ( A ⊔ B ) )
12		fodjuomni.p	. . . 4 |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
13	11, 12, 7	fodjuf 7204	. . 3 |- ( ph -> P e. ( 2o ^m O ) )
14	6, 10, 13	rspcdva 2869	. 2 |- ( ph -> ( E. w e. O ( P ` w ) = (/) \/ A. w e. O ( P ` w ) = 1o ) )
15	11	adantr 276	. . . . 5 |- ( ( ph /\ E. w e. O ( P ` w ) = (/) ) -> F : O -onto-> ( A ⊔ B ) )
16		simpr 110	. . . . . 6 |- ( ( ph /\ E. w e. O ( P ` w ) = (/) ) -> E. w e. O ( P ` w ) = (/) )
17		fveqeq2 5563	. . . . . . 7 |- ( w = v -> ( ( P ` w ) = (/) <-> ( P ` v ) = (/) ) )
18	17	cbvrexv 2727	. . . . . 6 |- ( E. w e. O ( P ` w ) = (/) <-> E. v e. O ( P ` v ) = (/) )
19	16, 18	sylib 122	. . . . 5 |- ( ( ph /\ E. w e. O ( P ` w ) = (/) ) -> E. v e. O ( P ` v ) = (/) )
20	15, 12, 19	fodjum 7205	. . . 4 |- ( ( ph /\ E. w e. O ( P ` w ) = (/) ) -> E. x x e. A )
21	20	ex 115	. . 3 |- ( ph -> ( E. w e. O ( P ` w ) = (/) -> E. x x e. A ) )
22	11	adantr 276	. . . . 5 |- ( ( ph /\ A. w e. O ( P ` w ) = 1o ) -> F : O -onto-> ( A ⊔ B ) )
23		simpr 110	. . . . 5 |- ( ( ph /\ A. w e. O ( P ` w ) = 1o ) -> A. w e. O ( P ` w ) = 1o )
24	22, 12, 23	fodju0 7206	. . . 4 |- ( ( ph /\ A. w e. O ( P ` w ) = 1o ) -> A = (/) )
25	24	ex 115	. . 3 |- ( ph -> ( A. w e. O ( P ` w ) = 1o -> A = (/) ) )
26	21, 25	orim12d 787	. 2 |- ( ph -> ( ( E. w e. O ( P ` w ) = (/) \/ A. w e. O ( P ` w ) = 1o ) -> ( E. x x e. A \/ A = (/) ) ) )
27	14, 26	mpd 13	1 |- ( ph -> ( E. x x e. A \/ A = (/) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   E.wrex 2473   (/)c0 3446   ifcif 3557   |-> cmpt 4090   -onto->wfo 5252   ` cfv 5254  (class class class)co 5918   1oc1o 6462   2oc2o 6463   ^m cmap 6702   ⊔ cdju 7096  inlcinl 7104  Omnicomni 7193

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-1o 6469  df-2o 6470  df-map 6704  df-dju 7097  df-inl 7106  df-inr 7107  df-omni 7194

This theorem is referenced by:  fodjuomni  7208