Theorem fodjuomnilemres 7346

Description: Lemma for fodjuomni 7347. 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 5638	. . . . . 6 |- ( f = P -> ( f ` w ) = ( P ` w ) )
2	1	eqeq1d 2240	. . . . 5 |- ( f = P -> ( ( f ` w ) = (/) <-> ( P ` w ) = (/) ) )
3	2	rexbidv 2533	. . . 4 |- ( f = P -> ( E. w e. O ( f ` w ) = (/) <-> E. w e. O ( P ` w ) = (/) ) )
4	1	eqeq1d 2240	. . . . 5 |- ( f = P -> ( ( f ` w ) = 1o <-> ( P ` w ) = 1o ) )
5	4	ralbidv 2532	. . . 4 |- ( f = P -> ( A. w e. O ( f ` w ) = 1o <-> A. w e. O ( P ` w ) = 1o ) )
6	3, 5	orbi12d 800	. . 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 7335	. . . . 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 7343	. . 3 |- ( ph -> P e. ( 2o ^m O ) )
14	6, 10, 13	rspcdva 2915	. 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 5648	. . . . . . 7 |- ( w = v -> ( ( P ` w ) = (/) <-> ( P ` v ) = (/) ) )
18	17	cbvrexv 2768	. . . . . 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 7344	. . . 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 7345	. . . 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 793	. 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 715   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   E.wrex 2511   (/)c0 3494   ifcif 3605   |-> cmpt 4150   -onto->wfo 5324   ` cfv 5326  (class class class)co 6017   1oc1o 6574   2oc2o 6575   ^m cmap 6816   ⊔ cdju 7235  inlcinl 7243  Omnicomni 7332

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-1o 6581  df-2o 6582  df-map 6818  df-dju 7236  df-inl 7245  df-inr 7246  df-omni 7333

This theorem is referenced by:  fodjuomni  7347