Theorem fodjumkv 7288

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 5599	. . . . . . 7 |- ( b = z -> (inl ` b ) = (inl ` z ) )
4	3	eqeq2d 2219	. . . . . 6 |- ( b = z -> ( ( F ` a ) = (inl ` b ) <-> ( F ` a ) = (inl ` z ) ) )
5	4	cbvrexv 2743	. . . . 5 |- ( E. b e. A ( F ` a ) = (inl ` b ) <-> E. z e. A ( F ` a ) = (inl ` z ) )
6		ifbi 3600	. . . . 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 4147	. . 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 5608	. . . . . 6 |- ( a = y -> ( ( F ` a ) = (inl ` z ) <-> ( F ` y ) = (inl ` z ) ) )
10	9	rexbidv 2509	. . . . 5 |- ( a = y -> ( E. z e. A ( F ` a ) = (inl ` z ) <-> E. z e. A ( F ` y ) = (inl ` z ) ) )
11	10	ifbid 3601	. . . 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 4156	. . 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 2228	. 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 7287	1 |- ( ph -> ( A =/= (/) -> E. x x e. A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   =/= wne 2378   E.wrex 2487   (/)c0 3468   ifcif 3579   |-> cmpt 4121   -onto->wfo 5288   ` cfv 5290   1oc1o 6518   ⊔ cdju 7165  inlcinl 7173  Markovcmarkov 7279

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-1o 6525  df-2o 6526  df-map 6760  df-dju 7166  df-inl 7175  df-inr 7176  df-markov 7280

This theorem is referenced by: (None)