Theorem fodjuomnilemres 7439

Description: Lemma for fodjuomni 7440. 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 5669	. . . . . 6 |- ( f = P -> ( f ` w ) = ( P ` w ) )
2	1	eqeq1d 2241	. . . . 5 |- ( f = P -> ( ( f ` w ) = (/) <-> ( P ` w ) = (/) ) )
3	2	rexbidv 2543	. . . 4 |- ( f = P -> ( E. w e. O ( f ` w ) = (/) <-> E. w e. O ( P ` w ) = (/) ) )
4	1	eqeq1d 2241	. . . . 5 |- ( f = P -> ( ( f ` w ) = 1o <-> ( P ` w ) = 1o ) )
5	4	ralbidv 2542	. . . 4 |- ( f = P -> ( A. w e. O ( f ` w ) = 1o <-> A. w e. O ( P ` w ) = 1o ) )
6	3, 5	orbi12d 801	. . 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 7428	. . . . 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 7436	. . 3 |- ( ph -> P e. ( 2o ^m O ) )
14	6, 10, 13	rspcdva 2926	. 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 5679	. . . . . . 7 |- ( w = v -> ( ( P ` w ) = (/) <-> ( P ` v ) = (/) ) )
18	17	cbvrexv 2779	. . . . . 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 7437	. . . 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 7438	. . . 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 794	. 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 716   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   E.wrex 2521   (/)c0 3508   ifcif 3620   |-> cmpt 4171   -onto->wfo 5350   ` cfv 5352  (class class class)co 6050   1oc1o 6640   2oc2o 6641   ^m cmap 6882   ⊔ cdju 7328  inlcinl 7336  Omnicomni 7425

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-1o 6647  df-2o 6648  df-map 6884  df-dju 7329  df-inl 7338  df-inr 7339  df-omni 7426

This theorem is referenced by:  fodjuomni  7440