Theorem fodjumkv 7453

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 5672	. . . . . . 7 |- ( b = z -> (inl ` b ) = (inl ` z ) )
4	3	eqeq2d 2246	. . . . . 6 |- ( b = z -> ( ( F ` a ) = (inl ` b ) <-> ( F ` a ) = (inl ` z ) ) )
5	4	cbvrexv 2781	. . . . 5 |- ( E. b e. A ( F ` a ) = (inl ` b ) <-> E. z e. A ( F ` a ) = (inl ` z ) )
6		ifbi 3645	. . . . 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 4199	. . 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 5681	. . . . . 6 |- ( a = y -> ( ( F ` a ) = (inl ` z ) <-> ( F ` y ) = (inl ` z ) ) )
10	9	rexbidv 2545	. . . . 5 |- ( a = y -> ( E. z e. A ( F ` a ) = (inl ` z ) <-> E. z e. A ( F ` y ) = (inl ` z ) ) )
11	10	ifbid 3646	. . . 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 4208	. . 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 2255	. 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 7452	1 |- ( ph -> ( A =/= (/) -> E. x x e. A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   =/= wne 2414   E.wrex 2523   (/)c0 3510   ifcif 3622   |-> cmpt 4173   -onto->wfo 5352   ` cfv 5354   1oc1o 6642   ⊔ cdju 7330  inlcinl 7338  Markovcmarkov 7444

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-2o 6650  df-map 6886  df-dju 7331  df-inl 7340  df-inr 7341  df-markov 7445

This theorem is referenced by: (None)