Theorem fodjumkv 7034

Description: A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)

Hypotheses

Ref	Expression
fodjumkv.o	|- ( ph -> M e. Markov )
fodjumkv.fo	|- ( ph -> F : M -onto-> ( A ⊔ B ) )

Assertion

Ref	Expression
fodjumkv	|- ( ph -> ( A =/= (/) -> E. x x e. A ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)   F( x)   M( x)

Proof of Theorem fodjumkv

Dummy variables a b y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjumkv.o	. 2 |- ( ph -> M e. Markov )
2		fodjumkv.fo	. 2 |- ( ph -> F : M -onto-> ( A ⊔ B ) )
3		fveq2 5421	. . . . . . 7 |- ( b = z -> (inl ` b ) = (inl ` z ) )
4	3	eqeq2d 2151	. . . . . 6 |- ( b = z -> ( ( F ` a ) = (inl ` b ) <-> ( F ` a ) = (inl ` z ) ) )
5	4	cbvrexv 2655	. . . . 5 |- ( E. b e. A ( F ` a ) = (inl ` b ) <-> E. z e. A ( F ` a ) = (inl ` z ) )
6		ifbi 3492	. . . . 5 |- ( ( E. b e. A ( F ` a ) = (inl ` b ) <-> E. z e. A ( F ` a ) = (inl ` z ) ) -> if ( E. b e. A ( F ` a ) = (inl ` b ) , (/) , 1o ) = if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o ) )
7	5, 6	ax-mp 5	. . . 4 |- if ( E. b e. A ( F ` a ) = (inl ` b ) , (/) , 1o ) = if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o )
8	7	mpteq2i 4015	. . 3 |- ( a e. M |-> if ( E. b e. A ( F ` a ) = (inl ` b ) , (/) , 1o ) ) = ( a e. M |-> if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o ) )
9		fveqeq2 5430	. . . . . 6 |- ( a = y -> ( ( F ` a ) = (inl ` z ) <-> ( F ` y ) = (inl ` z ) ) )
10	9	rexbidv 2438	. . . . 5 |- ( a = y -> ( E. z e. A ( F ` a ) = (inl ` z ) <-> E. z e. A ( F ` y ) = (inl ` z ) ) )
11	10	ifbid 3493	. . . 4 |- ( a = y -> if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o ) = if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
12	11	cbvmptv 4024	. . 3 |- ( a e. M |-> if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o ) ) = ( y e. M |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
13	8, 12	eqtri 2160	. 2 |- ( a e. M |-> if ( E. b e. A ( F ` a ) = (inl ` b ) , (/) , 1o ) ) = ( y e. M |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
14	1, 2, 13	fodjumkvlemres 7033	1 |- ( ph -> ( A =/= (/) -> E. x x e. A ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   =/= wne 2308   E.wrex 2417   (/)c0 3363   ifcif 3474   |-> cmpt 3989   -onto->wfo 5121   ` cfv 5123   1oc1o 6306   ⊔ cdju 6922  inlcinl 6930  Markovcmarkov 7025

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-1o 6313  df-2o 6314  df-map 6544  df-dju 6923  df-inl 6932  df-inr 6933  df-markov 7026

This theorem is referenced by: (None)