Theorem fodju0 7345

Description: Lemma for fodjuomni 7347 and fodjumkv 7358. A condition which shows that A is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)

Hypotheses

Ref	Expression
fodjuf.fo	|- ( ph -> F : O -onto-> ( A ⊔ B ) )
fodjuf.p	|- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
fodju0.1	|- ( ph -> A. w e. O ( P ` w ) = 1o )

Assertion

Ref	Expression
fodju0	|- ( ph -> A = (/) )

Distinct variable groups:   ph, y, z   y, O, z   z, A   z, B   z, F   y, A   y, F   w, O   w, P

Allowed substitution hints:   ph( w)   A( w)   B( y, w)   P( y, z)   F( w)

Proof of Theorem fodju0

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjuf.fo	. . . . 5 |- ( ph -> F : O -onto-> ( A ⊔ B ) )
2		djulcl 7249	. . . . 5 |- ( u e. A -> (inl ` u ) e. ( A ⊔ B ) )
3		foelrn 5892	. . . . 5 |- ( ( F : O -onto-> ( A ⊔ B ) /\ (inl ` u ) e. ( A ⊔ B ) ) -> E. v e. O (inl ` u ) = ( F ` v ) )
4	1, 2, 3	syl2an 289	. . . 4 |- ( ( ph /\ u e. A ) -> E. v e. O (inl ` u ) = ( F ` v ) )
5		fodjuf.p	. . . . . 6 |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
6		fveqeq2 5648	. . . . . . . 8 |- ( y = v -> ( ( F ` y ) = (inl ` z ) <-> ( F ` v ) = (inl ` z ) ) )
7	6	rexbidv 2533	. . . . . . 7 |- ( y = v -> ( E. z e. A ( F ` y ) = (inl ` z ) <-> E. z e. A ( F ` v ) = (inl ` z ) ) )
8	7	ifbid 3627	. . . . . 6 |- ( y = v -> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) = if ( E. z e. A ( F ` v ) = (inl ` z ) , (/) , 1o ) )
9		simprl 531	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> v e. O )
10		peano1 4692	. . . . . . . 8 |- (/) e. om
11	10	a1i 9	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (/) e. om )
12		1onn 6687	. . . . . . . 8 |- 1o e. om
13	12	a1i 9	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> 1o e. om )
14	1	fodjuomnilemdc 7342	. . . . . . . 8 |- ( ( ph /\ v e. O ) -> DECID E. z e. A ( F ` v ) = (inl ` z ) )
15	14	ad2ant2r 509	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> DECID E. z e. A ( F ` v ) = (inl ` z ) )
16	11, 13, 15	ifcldcd 3643	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> if ( E. z e. A ( F ` v ) = (inl ` z ) , (/) , 1o ) e. om )
17	5, 8, 9, 16	fvmptd3 5740	. . . . 5 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( P ` v ) = if ( E. z e. A ( F ` v ) = (inl ` z ) , (/) , 1o ) )
18		fveqeq2 5648	. . . . . 6 |- ( w = v -> ( ( P ` w ) = 1o <-> ( P ` v ) = 1o ) )
19		fodju0.1	. . . . . . 7 |- ( ph -> A. w e. O ( P ` w ) = 1o )
20	19	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> A. w e. O ( P ` w ) = 1o )
21	18, 20, 9	rspcdva 2915	. . . . 5 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( P ` v ) = 1o )
22		simplr 529	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> u e. A )
23		simprr 533	. . . . . . . 8 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (inl ` u ) = ( F ` v ) )
24	23	eqcomd 2237	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( F ` v ) = (inl ` u ) )
25		fveq2 5639	. . . . . . . 8 |- ( z = u -> (inl ` z ) = (inl ` u ) )
26	25	rspceeqv 2928	. . . . . . 7 |- ( ( u e. A /\ ( F ` v ) = (inl ` u ) ) -> E. z e. A ( F ` v ) = (inl ` z ) )
27	22, 24, 26	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> E. z e. A ( F ` v ) = (inl ` z ) )
28	27	iftrued 3612	. . . . 5 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> if ( E. z e. A ( F ` v ) = (inl ` z ) , (/) , 1o ) = (/) )
29	17, 21, 28	3eqtr3rd 2273	. . . 4 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (/) = 1o )
30	4, 29	rexlimddv 2655	. . 3 |- ( ( ph /\ u e. A ) -> (/) = 1o )
31		1n0 6599	. . . . 5 |- 1o =/= (/)
32	31	nesymi 2448	. . . 4 |- -. (/) = 1o
33	32	a1i 9	. . 3 |- ( ( ph /\ u e. A ) -> -. (/) = 1o )
34	30, 33	pm2.65da 667	. 2 |- ( ph -> -. u e. A )
35	34	eq0rdv 3539	1 |- ( ph -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 841   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   (/)c0 3494   ifcif 3605   |-> cmpt 4150   omcom 4688   -onto->wfo 5324   ` cfv 5326   1oc1o 6574   ⊔ cdju 7235  inlcinl 7243

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  fodjuomnilemres  7346  fodjumkvlemres  7357