Theorem fodju0 7135

Description: Lemma for fodjuomni 7137 and fodjumkv 7148. 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 7040	. . . . 5 |- ( u e. A -> (inl ` u ) e. ( A ⊔ B ) )
3		foelrn 5744	. . . . 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 5516	. . . . . . . 8 |- ( y = v -> ( ( F ` y ) = (inl ` z ) <-> ( F ` v ) = (inl ` z ) ) )
7	6	rexbidv 2476	. . . . . . 7 |- ( y = v -> ( E. z e. A ( F ` y ) = (inl ` z ) <-> E. z e. A ( F ` v ) = (inl ` z ) ) )
8	7	ifbid 3553	. . . . . 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 529	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> v e. O )
10		peano1 4587	. . . . . . . 8 |- (/) e. om
11	10	a1i 9	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (/) e. om )
12		1onn 6511	. . . . . . . 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 7132	. . . . . . . 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 3567	. . . . . 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 5601	. . . . 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 5516	. . . . . 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 2844	. . . . 5 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( P ` v ) = 1o )
22		simplr 528	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> u e. A )
23		simprr 531	. . . . . . . 8 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (inl ` u ) = ( F ` v ) )
24	23	eqcomd 2181	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( F ` v ) = (inl ` u ) )
25		fveq2 5507	. . . . . . . 8 |- ( z = u -> (inl ` z ) = (inl ` u ) )
26	25	rspceeqv 2857	. . . . . . 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 3539	. . . . 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 2217	. . . 4 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (/) = 1o )
30	4, 29	rexlimddv 2597	. . 3 |- ( ( ph /\ u e. A ) -> (/) = 1o )
31		1n0 6423	. . . . 5 |- 1o =/= (/)
32	31	nesymi 2391	. . . 4 |- -. (/) = 1o
33	32	a1i 9	. . 3 |- ( ( ph /\ u e. A ) -> -. (/) = 1o )
34	30, 33	pm2.65da 661	. 2 |- ( ph -> -. u e. A )
35	34	eq0rdv 3465	1 |- ( ph -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 834   = wceq 1353   e. wcel 2146   A.wral 2453   E.wrex 2454   (/)c0 3420   ifcif 3532   |-> cmpt 4059   omcom 4583   -onto->wfo 5206   ` cfv 5208   1oc1o 6400   ⊔ cdju 7026  inlcinl 7034

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-dju 7027  df-inl 7036  df-inr 7037

This theorem is referenced by:  fodjuomnilemres  7136  fodjumkvlemres  7147