Theorem fodjuomnilemdc 7203

Description: Lemma for fodjuomni 7208. 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 5476	. . . . . 6 |- ( F : O -onto-> ( A ⊔ B ) -> F : O --> ( A ⊔ B ) )
3	1, 2	syl 14	. . . . 5 |- ( ph -> F : O --> ( A ⊔ B ) )
4	3	ffvelcdmda 5693	. . . 4 |- ( ( ph /\ X e. O ) -> ( F ` X ) e. ( A ⊔ B ) )
5		djur 7128	. . . 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 1539	. . . . . . . 8 |- F/ z ( ph /\ X e. O )
8		nfre1 2537	. . . . . . . 8 |- F/ z E. z e. B ( F ` X ) = (inr ` z )
9	7, 8	nfan 1576	. . . . . . 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 5554	. . . . . . . . . . . 12 |- ( z = w -> (inr ` z ) = (inr ` w ) )
12	11	eqeq2d 2205	. . . . . . . . . . 11 |- ( z = w -> ( ( F ` X ) = (inr ` z ) <-> ( F ` X ) = (inr ` w ) ) )
13	12	cbvrexv 2727	. . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 |- z e. _V
16		vex 2763	. . . . . . . . . . . . . . 15 |- w e. _V
17		djune 7137	. . . . . . . . . . . . . . 15 |- ( ( z e. _V /\ w e. _V ) -> (inl ` z ) =/= (inr ` w ) )
18	15, 16, 17	mp2an 426	. . . . . . . . . . . . . 14 |- (inl ` z ) =/= (inr ` w )
19		neeq2 2378	. . . . . . . . . . . . . 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 2450	. . . . . . . . . . . 12 |- ( ( F ` X ) = (inr ` w ) -> ( F ` X ) =/= (inl ` z ) )
22	21	neneqd 2385	. . . . . . . . . . 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 2615	. . . . . . . . 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 2565	. . . . . 6 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> A. z e. A -. ( F ` X ) = (inl ` z ) )
28		ralnex 2482	. . . . . 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 1364   e. wcel 2164   =/= wne 2364   A.wral 2472   E.wrex 2473   _Vcvv 2760   -->wf 5250   -onto->wfo 5252   ` cfv 5254   ⊔ cdju 7096  inlcinl 7104  inrcinr 7105

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-dju 7097  df-inl 7106  df-inr 7107

This theorem is referenced by:  fodjuf  7204  fodjum  7205  fodju0  7206