Theorem fodjum 7122

Description: Lemma for fodjuomni 7125 and fodjumkv 7136. 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 6411	. . . . . . . . 9 |- 1o =/= (/)
3	2	nesymi 2386	. . . . . . . 8 |- -. (/) = 1o
4	3	intnan 924	. . . . . . 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 527	. . . . . . . 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 5505	. . . . . . . . . . 11 |- ( y = w -> ( ( F ` y ) = (inl ` z ) <-> ( F ` w ) = (inl ` z ) ) )
9	8	rexbidv 2471	. . . . . . . . . 10 |- ( y = w -> ( E. z e. A ( F ` y ) = (inl ` z ) <-> E. z e. A ( F ` w ) = (inl ` z ) ) )
10	9	ifbid 3547	. . . . . . . . 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 526	. . . . . . . . 9 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> w e. O )
12		peano1 4578	. . . . . . . . . . 11 |- (/) e. om
13	12	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) e. om )
14		1onn 6499	. . . . . . . . . . 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 7120	. . . . . . . . . . 11 |- ( ( ph /\ w e. O ) -> DECID E. z e. A ( F ` w ) = (inl ` z ) )
18	17	adantrr 476	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> DECID E. z e. A ( F ` w ) = (inl ` z ) )
19	13, 15, 18	ifcldcd 3561	. . . . . . . . 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 5589	. . . . . . . 8 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( P ` w ) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
21	6, 20	eqtr3d 2205	. . . . . . 7 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
22		eqifdc 3560	. . . . . . . 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 146	. . . . . 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 1344	. . . . 5 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = (/) ) )
26	25	simpld 111	. . . 4 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> E. z e. A ( F ` w ) = (inl ` z ) )
27		rexm 3514	. . . 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 2231	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
30	29	cbvexv 1911	. . 3 |- ( E. z z e. A <-> E. x x e. A )
31	28, 30	sylib 121	. 2 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> E. x x e. A )
32	1, 31	rexlimddv 2592	1 |- ( ph -> E. x x e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   = wceq 1348   E.wex 1485   e. wcel 2141   E.wrex 2449   (/)c0 3414   ifcif 3526   |-> cmpt 4050   omcom 4574   -onto->wfo 5196   ` cfv 5198   1oc1o 6388   ⊔ cdju 7014  inlcinl 7022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025

This theorem is referenced by:  fodjuomnilemres  7124  fodjumkvlemres  7135