Theorem fodjum 7207

Description: Lemma for fodjuomni 7210 and fodjumkv 7221. A condition which shows that A is inhabited. (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 ) )
fodjum.z	|- ( ph -> E. w e. O ( P ` w ) = (/) )

Assertion

Ref	Expression
fodjum	|- ( ph -> E. x x e. A )

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

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

Proof of Theorem fodjum

Step	Hyp	Ref	Expression
1		fodjum.z	. 2 |- ( ph -> E. w e. O ( P ` w ) = (/) )
2		1n0 6487	. . . . . . . . 9 |- 1o =/= (/)
3	2	nesymi 2410	. . . . . . . 8 |- -. (/) = 1o
4	3	intnan 930	. . . . . . 7 |- -. ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o )
5	4	a1i 9	. . . . . 6 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> -. ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o ) )
6		simprr 531	. . . . . . . 8 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( P ` w ) = (/) )
7		fodjuf.p	. . . . . . . . 9 |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
8		fveqeq2 5564	. . . . . . . . . . 11 |- ( y = w -> ( ( F ` y ) = (inl ` z ) <-> ( F ` w ) = (inl ` z ) ) )
9	8	rexbidv 2495	. . . . . . . . . 10 |- ( y = w -> ( E. z e. A ( F ` y ) = (inl ` z ) <-> E. z e. A ( F ` w ) = (inl ` z ) ) )
10	9	ifbid 3579	. . . . . . . . 9 |- ( y = w -> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
11		simprl 529	. . . . . . . . 9 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> w e. O )
12		peano1 4627	. . . . . . . . . . 11 |- (/) e. om
13	12	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) e. om )
14		1onn 6575	. . . . . . . . . . 11 |- 1o e. om
15	14	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> 1o e. om )
16		fodjuf.fo	. . . . . . . . . . . 12 |- ( ph -> F : O -onto-> ( A ⊔ B ) )
17	16	fodjuomnilemdc 7205	. . . . . . . . . . 11 |- ( ( ph /\ w e. O ) -> DECID E. z e. A ( F ` w ) = (inl ` z ) )
18	17	adantrr 479	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> DECID E. z e. A ( F ` w ) = (inl ` z ) )
19	13, 15, 18	ifcldcd 3594	. . . . . . . . 9 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) e. om )
20	7, 10, 11, 19	fvmptd3 5652	. . . . . . . 8 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( P ` w ) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
21	6, 20	eqtr3d 2228	. . . . . . 7 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
22		eqifdc 3593	. . . . . . . 8 |- (DECID E. z e. A ( F ` w ) = (inl ` z ) -> ( (/) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) <-> ( ( E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = (/) ) \/ ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o ) ) ) )
23	18, 22	syl 14	. . . . . . 7 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( (/) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) <-> ( ( E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = (/) ) \/ ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o ) ) ) )
24	21, 23	mpbid 147	. . . . . 6 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( ( E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = (/) ) \/ ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o ) ) )
25	5, 24	ecased 1360	. . . . 5 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = (/) ) )
26	25	simpld 112	. . . 4 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> E. z e. A ( F ` w ) = (inl ` z ) )
27		rexm 3547	. . . 4 |- ( E. z e. A ( F ` w ) = (inl ` z ) -> E. z z e. A )
28	26, 27	syl 14	. . 3 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> E. z z e. A )
29		eleq1w 2254	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
30	29	cbvexv 1930	. . 3 |- ( E. z z e. A <-> E. x x e. A )
31	28, 30	sylib 122	. 2 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> E. x x e. A )
32	1, 31	rexlimddv 2616	1 |- ( ph -> E. x x e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   E.wex 1503   e. wcel 2164   E.wrex 2473   (/)c0 3447   ifcif 3558   |-> cmpt 4091   omcom 4623   -onto->wfo 5253   ` cfv 5255   1oc1o 6464   ⊔ cdju 7098  inlcinl 7106

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 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-1o 6471  df-dju 7099  df-inl 7108  df-inr 7109

This theorem is referenced by:  fodjuomnilemres  7209  fodjumkvlemres  7220