Theorem fodjuomnilemdc 7108

Description: Lemma for fodjuomni 7113. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.)

Hypothesis

Ref	Expression
fodjuomnilemdc.fo	|- ( ph -> F : O -onto-> ( A ⊔ B ) )

Assertion

Ref	Expression
fodjuomnilemdc	|- ( ( ph /\ X e. O ) -> DECID E. z e. A ( F ` X ) = (inl ` z ) )

Distinct variable groups:   z, A   z, B   z, F   z, O   z, X   ph, z

Proof of Theorem fodjuomnilemdc

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjuomnilemdc.fo	. . . . . 6 |- ( ph -> F : O -onto-> ( A ⊔ B ) )
2		fof 5410	. . . . . 6 |- ( F : O -onto-> ( A ⊔ B ) -> F : O --> ( A ⊔ B ) )
3	1, 2	syl 14	. . . . 5 |- ( ph -> F : O --> ( A ⊔ B ) )
4	3	ffvelrnda 5620	. . . 4 |- ( ( ph /\ X e. O ) -> ( F ` X ) e. ( A ⊔ B ) )
5		djur 7034	. . . 4 |- ( ( F ` X ) e. ( A ⊔ B ) <-> ( E. z e. A ( F ` X ) = (inl ` z ) \/ E. z e. B ( F ` X ) = (inr ` z ) ) )
6	4, 5	sylib 121	. . 3 |- ( ( ph /\ X e. O ) -> ( E. z e. A ( F ` X ) = (inl ` z ) \/ E. z e. B ( F ` X ) = (inr ` z ) ) )
7		nfv 1516	. . . . . . . 8 |- F/ z ( ph /\ X e. O )
8		nfre1 2509	. . . . . . . 8 |- F/ z E. z e. B ( F ` X ) = (inr ` z )
9	7, 8	nfan 1553	. . . . . . 7 |- F/ z ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) )
10		simpr 109	. . . . . . . . . 10 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> E. z e. B ( F ` X ) = (inr ` z ) )
11		fveq2 5486	. . . . . . . . . . . 12 |- ( z = w -> (inr ` z ) = (inr ` w ) )
12	11	eqeq2d 2177	. . . . . . . . . . 11 |- ( z = w -> ( ( F ` X ) = (inr ` z ) <-> ( F ` X ) = (inr ` w ) ) )
13	12	cbvrexv 2693	. . . . . . . . . 10 |- ( E. z e. B ( F ` X ) = (inr ` z ) <-> E. w e. B ( F ` X ) = (inr ` w ) )
14	10, 13	sylib 121	. . . . . . . . 9 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> E. w e. B ( F ` X ) = (inr ` w ) )
15		vex 2729	. . . . . . . . . . . . . . 15 |- z e. _V
16		vex 2729	. . . . . . . . . . . . . . 15 |- w e. _V
17		djune 7043	. . . . . . . . . . . . . . 15 |- ( ( z e. _V /\ w e. _V ) -> (inl ` z ) =/= (inr ` w ) )
18	15, 16, 17	mp2an 423	. . . . . . . . . . . . . 14 |- (inl ` z ) =/= (inr ` w )
19		neeq2 2350	. . . . . . . . . . . . . 14 |- ( ( F ` X ) = (inr ` w ) -> ( (inl ` z ) =/= ( F ` X ) <-> (inl ` z ) =/= (inr ` w ) ) )
20	18, 19	mpbiri 167	. . . . . . . . . . . . 13 |- ( ( F ` X ) = (inr ` w ) -> (inl ` z ) =/= ( F ` X ) )
21	20	necomd 2422	. . . . . . . . . . . 12 |- ( ( F ` X ) = (inr ` w ) -> ( F ` X ) =/= (inl ` z ) )
22	21	neneqd 2357	. . . . . . . . . . 11 |- ( ( F ` X ) = (inr ` w ) -> -. ( F ` X ) = (inl ` z ) )
23	22	a1i 9	. . . . . . . . . 10 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> ( ( F ` X ) = (inr ` w ) -> -. ( F ` X ) = (inl ` z ) ) )
24	23	rexlimdvw 2587	. . . . . . . . 9 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> ( E. w e. B ( F ` X ) = (inr ` w ) -> -. ( F ` X ) = (inl ` z ) ) )
25	14, 24	mpd 13	. . . . . . . 8 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> -. ( F ` X ) = (inl ` z ) )
26	25	a1d 22	. . . . . . 7 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> ( z e. A -> -. ( F ` X ) = (inl ` z ) ) )
27	9, 26	ralrimi 2537	. . . . . 6 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> A. z e. A -. ( F ` X ) = (inl ` z ) )
28		ralnex 2454	. . . . . 6 |- ( A. z e. A -. ( F ` X ) = (inl ` z ) <-> -. E. z e. A ( F ` X ) = (inl ` z ) )
29	27, 28	sylib 121	. . . . 5 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> -. E. z e. A ( F ` X ) = (inl ` z ) )
30	29	ex 114	. . . 4 |- ( ( ph /\ X e. O ) -> ( E. z e. B ( F ` X ) = (inr ` z ) -> -. E. z e. A ( F ` X ) = (inl ` z ) ) )
31	30	orim2d 778	. . 3 |- ( ( ph /\ X e. O ) -> ( ( E. z e. A ( F ` X ) = (inl ` z ) \/ E. z e. B ( F ` X ) = (inr ` z ) ) -> ( E. z e. A ( F ` X ) = (inl ` z ) \/ -. E. z e. A ( F ` X ) = (inl ` z ) ) ) )
32	6, 31	mpd 13	. 2 |- ( ( ph /\ X e. O ) -> ( E. z e. A ( F ` X ) = (inl ` z ) \/ -. E. z e. A ( F ` X ) = (inl ` z ) ) )
33		df-dc 825	. 2 |- (DECID E. z e. A ( F ` X ) = (inl ` z ) <-> ( E. z e. A ( F ` X ) = (inl ` z ) \/ -. E. z e. A ( F ` X ) = (inl ` z ) ) )
34	32, 33	sylibr 133	1 |- ( ( ph /\ X e. O ) -> DECID E. z e. A ( F ` X ) = (inl ` z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 824   = wceq 1343   e. wcel 2136   =/= wne 2336   A.wral 2444   E.wrex 2445   _Vcvv 2726   -->wf 5184   -onto->wfo 5186   ` cfv 5188   ⊔ cdju 7002  inlcinl 7010  inrcinr 7011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013

This theorem is referenced by:  fodjuf  7109  fodjum  7110  fodju0  7111