Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  fodjumkv GIF version

Theorem fodjumkv 6984
 Description: A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
Hypotheses
Ref Expression
fodjumkv.o (𝜑𝑀 ∈ Markov)
fodjumkv.fo (𝜑𝐹:𝑀onto→(𝐴𝐵))
Assertion
Ref Expression
fodjumkv (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(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.
StepHypRef Expression
1 fodjumkv.o . 2 (𝜑𝑀 ∈ Markov)
2 fodjumkv.fo . 2 (𝜑𝐹:𝑀onto→(𝐴𝐵))
3 fveq2 5375 . . . . . . 7 (𝑏 = 𝑧 → (inl‘𝑏) = (inl‘𝑧))
43eqeq2d 2126 . . . . . 6 (𝑏 = 𝑧 → ((𝐹𝑎) = (inl‘𝑏) ↔ (𝐹𝑎) = (inl‘𝑧)))
54cbvrexv 2629 . . . . 5 (∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏) ↔ ∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧))
6 ifbi 3458 . . . . 5 ((∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏) ↔ ∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧)) → if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o))
75, 6ax-mp 7 . . . 4 if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o)
87mpteq2i 3975 . . 3 (𝑎𝑀 ↦ if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑎𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o))
9 fveqeq2 5384 . . . . . 6 (𝑎 = 𝑦 → ((𝐹𝑎) = (inl‘𝑧) ↔ (𝐹𝑦) = (inl‘𝑧)))
109rexbidv 2412 . . . . 5 (𝑎 = 𝑦 → (∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧) ↔ ∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧)))
1110ifbid 3459 . . . 4 (𝑎 = 𝑦 → if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
1211cbvmptv 3984 . . 3 (𝑎𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o)) = (𝑦𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
138, 12eqtri 2135 . 2 (𝑎𝑀 ↦ if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑦𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
141, 2, 13fodjumkvlemres 6983 1 (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1314  ∃wex 1451   ∈ wcel 1463   ≠ wne 2282  ∃wrex 2391  ∅c0 3329  ifcif 3440   ↦ cmpt 3949  –onto→wfo 5079  ‘cfv 5081  1oc1o 6260   ⊔ cdju 6874  inlcinl 6882  Markovcmarkov 6975 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412 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 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-1o 6267  df-2o 6268  df-map 6498  df-dju 6875  df-inl 6884  df-inr 6885  df-markov 6976 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator