Theorem fodjuomnilemdc 7245

Description: Lemma for fodjuomni 7250. 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 5497	. . . . . 6 |- ( F : O -onto-> ( A ⊔ B ) -> F : O --> ( A ⊔ B ) )
3	1, 2	syl 14	. . . . 5 |- ( ph -> F : O --> ( A ⊔ B ) )
4	3	ffvelcdmda 5714	. . . 4 |- ( ( ph /\ X e. O ) -> ( F ` X ) e. ( A ⊔ B ) )
5		djur 7170	. . . 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 122	. . 3 |- ( ( ph /\ X e. O ) -> ( E. z e. A ( F ` X ) = (inl ` z ) \/ E. z e. B ( F ` X ) = (inr ` z ) ) )
7		nfv 1550	. . . . . . . 8 |- F/ z ( ph /\ X e. O )
8		nfre1 2548	. . . . . . . 8 |- F/ z E. z e. B ( F ` X ) = (inr ` z )
9	7, 8	nfan 1587	. . . . . . 7 |- F/ z ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) )
10		simpr 110	. . . . . . . . . 10 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> E. z e. B ( F ` X ) = (inr ` z ) )
11		fveq2 5575	. . . . . . . . . . . 12 |- ( z = w -> (inr ` z ) = (inr ` w ) )
12	11	eqeq2d 2216	. . . . . . . . . . 11 |- ( z = w -> ( ( F ` X ) = (inr ` z ) <-> ( F ` X ) = (inr ` w ) ) )
13	12	cbvrexv 2738	. . . . . . . . . 10 |- ( E. z e. B ( F ` X ) = (inr ` z ) <-> E. w e. B ( F ` X ) = (inr ` w ) )
14	10, 13	sylib 122	. . . . . . . . 9 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> E. w e. B ( F ` X ) = (inr ` w ) )
15		vex 2774	. . . . . . . . . . . . . . 15 |- z e. _V
16		vex 2774	. . . . . . . . . . . . . . 15 |- w e. _V
17		djune 7179	. . . . . . . . . . . . . . 15 |- ( ( z e. _V /\ w e. _V ) -> (inl ` z ) =/= (inr ` w ) )
18	15, 16, 17	mp2an 426	. . . . . . . . . . . . . 14 |- (inl ` z ) =/= (inr ` w )
19		neeq2 2389	. . . . . . . . . . . . . 14 |- ( ( F ` X ) = (inr ` w ) -> ( (inl ` z ) =/= ( F ` X ) <-> (inl ` z ) =/= (inr ` w ) ) )
20	18, 19	mpbiri 168	. . . . . . . . . . . . 13 |- ( ( F ` X ) = (inr ` w ) -> (inl ` z ) =/= ( F ` X ) )
21	20	necomd 2461	. . . . . . . . . . . 12 |- ( ( F ` X ) = (inr ` w ) -> ( F ` X ) =/= (inl ` z ) )
22	21	neneqd 2396	. . . . . . . . . . 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 2626	. . . . . . . . 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 2576	. . . . . 6 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> A. z e. A -. ( F ` X ) = (inl ` z ) )
28		ralnex 2493	. . . . . 6 |- ( A. z e. A -. ( F ` X ) = (inl ` z ) <-> -. E. z e. A ( F ` X ) = (inl ` z ) )
29	27, 28	sylib 122	. . . . 5 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> -. E. z e. A ( F ` X ) = (inl ` z ) )
30	29	ex 115	. . . 4 |- ( ( ph /\ X e. O ) -> ( E. z e. B ( F ` X ) = (inr ` z ) -> -. E. z e. A ( F ` X ) = (inl ` z ) ) )
31	30	orim2d 789	. . 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 836	. 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 134	1 |- ( ( ph /\ X e. O ) -> DECID E. z e. A ( F ` X ) = (inl ` z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1372   e. wcel 2175   =/= wne 2375   A.wral 2483   E.wrex 2484   _Vcvv 2771   -->wf 5266   -onto->wfo 5268   ` cfv 5270   ⊔ cdju 7138  inlcinl 7146  inrcinr 7147

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1st 6225  df-2nd 6226  df-1o 6501  df-dju 7139  df-inl 7148  df-inr 7149

This theorem is referenced by:  fodjuf  7246  fodjum  7247  fodju0  7248