Theorem fodjuomnilemdc 7448

Description: Lemma for fodjuomni 7453. 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 5595	. . . . . 6 |- ( F : O -onto-> ( A ⊔ B ) -> F : O --> ( A ⊔ B ) )
3	1, 2	syl 14	. . . . 5 |- ( ph -> F : O --> ( A ⊔ B ) )
4	3	ffvelcdmda 5817	. . . 4 |- ( ( ph /\ X e. O ) -> ( F ` X ) e. ( A ⊔ B ) )
5		djur 7373	. . . 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 1577	. . . . . . . 8 |- F/ z ( ph /\ X e. O )
8		nfre1 2587	. . . . . . . 8 |- F/ z E. z e. B ( F ` X ) = (inr ` z )
9	7, 8	nfan 1614	. . . . . . 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 5675	. . . . . . . . . . . 12 |- ( z = w -> (inr ` z ) = (inr ` w ) )
12	11	eqeq2d 2246	. . . . . . . . . . 11 |- ( z = w -> ( ( F ` X ) = (inr ` z ) <-> ( F ` X ) = (inr ` w ) ) )
13	12	cbvrexv 2781	. . . . . . . . . 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 2818	. . . . . . . . . . . . . . 15 |- z e. _V
16		vex 2818	. . . . . . . . . . . . . . 15 |- w e. _V
17		djune 7382	. . . . . . . . . . . . . . 15 |- ( ( z e. _V /\ w e. _V ) -> (inl ` z ) =/= (inr ` w ) )
18	15, 16, 17	mp2an 426	. . . . . . . . . . . . . 14 |- (inl ` z ) =/= (inr ` w )
19		neeq2 2428	. . . . . . . . . . . . . 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 2500	. . . . . . . . . . . 12 |- ( ( F ` X ) = (inr ` w ) -> ( F ` X ) =/= (inl ` z ) )
22	21	neneqd 2435	. . . . . . . . . . 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 2666	. . . . . . . . 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 2615	. . . . . 6 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> A. z e. A -. ( F ` X ) = (inl ` z ) )
28		ralnex 2532	. . . . . 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 796	. . 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 843	. 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 716  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   E.wrex 2523   _Vcvv 2815   -->wf 5353   -onto->wfo 5355   ` cfv 5357   ⊔ cdju 7341  inlcinl 7349  inrcinr 7350

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-dju 7342  df-inl 7351  df-inr 7352

This theorem is referenced by:  fodjuf  7449  fodjum  7450  fodju0  7451