Theorem fodjuomnilemres 7148

Description: Lemma for fodjuomni 7149. 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 5516	. . . . . 6 |- ( f = P -> ( f ` w ) = ( P ` w ) )
2	1	eqeq1d 2186	. . . . 5 |- ( f = P -> ( ( f ` w ) = (/) <-> ( P ` w ) = (/) ) )
3	2	rexbidv 2478	. . . 4 |- ( f = P -> ( E. w e. O ( f ` w ) = (/) <-> E. w e. O ( P ` w ) = (/) ) )
4	1	eqeq1d 2186	. . . . 5 |- ( f = P -> ( ( f ` w ) = 1o <-> ( P ` w ) = 1o ) )
5	4	ralbidv 2477	. . . 4 |- ( f = P -> ( A. w e. O ( f ` w ) = 1o <-> A. w e. O ( P ` w ) = 1o ) )
6	3, 5	orbi12d 793	. . 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 7137	. . . . 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 7145	. . 3 |- ( ph -> P e. ( 2o ^m O ) )
14	6, 10, 13	rspcdva 2848	. 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 5526	. . . . . . 7 |- ( w = v -> ( ( P ` w ) = (/) <-> ( P ` v ) = (/) ) )
18	17	cbvrexv 2706	. . . . . 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 7146	. . . 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 7147	. . . 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 786	. 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 708   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   E.wrex 2456   (/)c0 3424   ifcif 3536   |-> cmpt 4066   -onto->wfo 5216   ` cfv 5218  (class class class)co 5877   1oc1o 6412   2oc2o 6413   ^m cmap 6650   ⊔ cdju 7038  inlcinl 7046  Omnicomni 7134

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 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-1o 6419  df-2o 6420  df-map 6652  df-dju 7039  df-inl 7048  df-inr 7049  df-omni 7135

This theorem is referenced by:  fodjuomni  7149