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Theorem fodjumkv 7106
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 5471 . . . . . . 7 (𝑏 = 𝑧 → (inl‘𝑏) = (inl‘𝑧))
43eqeq2d 2169 . . . . . 6 (𝑏 = 𝑧 → ((𝐹𝑎) = (inl‘𝑏) ↔ (𝐹𝑎) = (inl‘𝑧)))
54cbvrexv 2681 . . . . 5 (∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏) ↔ ∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧))
6 ifbi 3526 . . . . 5 ((∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏) ↔ ∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧)) → if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o))
75, 6ax-mp 5 . . . 4 if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o)
87mpteq2i 4054 . . 3 (𝑎𝑀 ↦ if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑎𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o))
9 fveqeq2 5480 . . . . . 6 (𝑎 = 𝑦 → ((𝐹𝑎) = (inl‘𝑧) ↔ (𝐹𝑦) = (inl‘𝑧)))
109rexbidv 2458 . . . . 5 (𝑎 = 𝑦 → (∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧) ↔ ∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧)))
1110ifbid 3527 . . . 4 (𝑎 = 𝑦 → if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
1211cbvmptv 4063 . . 3 (𝑎𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o)) = (𝑦𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
138, 12eqtri 2178 . 2 (𝑎𝑀 ↦ if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑦𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
141, 2, 13fodjumkvlemres 7105 1 (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1335  wex 1472  wcel 2128  wne 2327  wrex 2436  c0 3395  ifcif 3506  cmpt 4028  ontowfo 5171  cfv 5173  1oc1o 6359  cdju 6984  inlcinl 6992  Markovcmarkov 7097
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-iord 4329  df-on 4331  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-1o 6366  df-2o 6367  df-map 6598  df-dju 6985  df-inl 6994  df-inr 6995  df-markov 7098
This theorem is referenced by: (None)
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