Theorem fodjuomnilemdc 7435

Description: Lemma for fodjuomni 7440. 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 5590	. . . . . 6 |- ( F : O -onto-> ( A ⊔ B ) -> F : O --> ( A ⊔ B ) )
3	1, 2	syl 14	. . . . 5 |- ( ph -> F : O --> ( A ⊔ B ) )
4	3	ffvelcdmda 5812	. . . 4 |- ( ( ph /\ X e. O ) -> ( F ` X ) e. ( A ⊔ B ) )
5		djur 7360	. . . 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 2585	. . . . . . . 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 5670	. . . . . . . . . . . 12 |- ( z = w -> (inr ` z ) = (inr ` w ) )
12	11	eqeq2d 2244	. . . . . . . . . . 11 |- ( z = w -> ( ( F ` X ) = (inr ` z ) <-> ( F ` X ) = (inr ` w ) ) )
13	12	cbvrexv 2779	. . . . . . . . . 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 2816	. . . . . . . . . . . . . . 15 |- z e. _V
16		vex 2816	. . . . . . . . . . . . . . 15 |- w e. _V
17		djune 7369	. . . . . . . . . . . . . . 15 |- ( ( z e. _V /\ w e. _V ) -> (inl ` z ) =/= (inr ` w ) )
18	15, 16, 17	mp2an 426	. . . . . . . . . . . . . 14 |- (inl ` z ) =/= (inr ` w )
19		neeq2 2426	. . . . . . . . . . . . . 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 2498	. . . . . . . . . . . 12 |- ( ( F ` X ) = (inr ` w ) -> ( F ` X ) =/= (inl ` z ) )
22	21	neneqd 2433	. . . . . . . . . . 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 2664	. . . . . . . . 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 2613	. . . . . 6 |- ( ( ( ph /\ X e. O ) /\ E. z e. B ( F ` X ) = (inr ` z ) ) -> A. z e. A -. ( F ` X ) = (inl ` z ) )
28		ralnex 2530	. . . . . 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 2203   =/= wne 2412   A.wral 2520   E.wrex 2521   _Vcvv 2813   -->wf 5348   -onto->wfo 5350   ` cfv 5352   ⊔ cdju 7328  inlcinl 7336  inrcinr 7337

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1st 6334  df-2nd 6335  df-1o 6647  df-dju 7329  df-inl 7338  df-inr 7339

This theorem is referenced by:  fodjuf  7436  fodjum  7437  fodju0  7438