Theorem fodjuomnilemres 7315

Description: Lemma for fodjuomni 7316. 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 5626	. . . . . 6 |- ( f = P -> ( f ` w ) = ( P ` w ) )
2	1	eqeq1d 2238	. . . . 5 |- ( f = P -> ( ( f ` w ) = (/) <-> ( P ` w ) = (/) ) )
3	2	rexbidv 2531	. . . 4 |- ( f = P -> ( E. w e. O ( f ` w ) = (/) <-> E. w e. O ( P ` w ) = (/) ) )
4	1	eqeq1d 2238	. . . . 5 |- ( f = P -> ( ( f ` w ) = 1o <-> ( P ` w ) = 1o ) )
5	4	ralbidv 2530	. . . 4 |- ( f = P -> ( A. w e. O ( f ` w ) = 1o <-> A. w e. O ( P ` w ) = 1o ) )
6	3, 5	orbi12d 798	. . 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 7304	. . . . 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 7312	. . 3 |- ( ph -> P e. ( 2o ^m O ) )
14	6, 10, 13	rspcdva 2912	. 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 5636	. . . . . . 7 |- ( w = v -> ( ( P ` w ) = (/) <-> ( P ` v ) = (/) ) )
18	17	cbvrexv 2766	. . . . . 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 7313	. . . 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 7314	. . . 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 791	. 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 713   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   (/)c0 3491   ifcif 3602   |-> cmpt 4145   -onto->wfo 5316   ` cfv 5318  (class class class)co 6001   1oc1o 6555   2oc2o 6556   ^m cmap 6795   ⊔ cdju 7204  inlcinl 7212  Omnicomni 7301

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-1o 6562  df-2o 6563  df-map 6797  df-dju 7205  df-inl 7214  df-inr 7215  df-omni 7302

This theorem is referenced by:  fodjuomni  7316