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Theorem fodjuomni 6987
 Description: A condition which ensures is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)
Hypotheses
Ref Expression
fodjuomni.o Omni
fodjuomni.fo
Assertion
Ref Expression
fodjuomni
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem fodjuomni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjuomni.o . 2 Omni
2 fodjuomni.fo . 2
3 fveq2 5387 . . . . . . 7 inl inl
43eqeq2d 2127 . . . . . 6 inl inl
54cbvrexv 2630 . . . . 5 inl inl
6 ifbi 3460 . . . . 5 inl inl inl inl
75, 6ax-mp 5 . . . 4 inl inl
87mpteq2i 3983 . . 3 inl inl
9 fveq2 5387 . . . . . . 7
109eqeq1d 2124 . . . . . 6 inl inl
1110rexbidv 2413 . . . . 5 inl inl
1211ifbid 3461 . . . 4 inl inl
1312cbvmptv 3992 . . 3 inl inl
148, 13eqtri 2136 . 2 inl inl
151, 2, 14fodjuomnilemres 6986 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wo 680   wceq 1314  wex 1451   wcel 1463  wrex 2392  c0 3331  cif 3442   cmpt 3957  wfo 5089  cfv 5091  c1o 6272   ⊔ cdju 6888  inlcinl 6896  Omnicomni 6970 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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420 This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-1o 6279  df-2o 6280  df-map 6510  df-dju 6889  df-inl 6898  df-inr 6899  df-omni 6972 This theorem is referenced by:  ctssexmid  6990  exmidunben  11845  exmidsbthrlem  13051  sbthomlem  13054
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