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

Theorem tfrlemiex 6357
Description: Lemma for tfrlemi1 6358. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlemisucfn.2 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
tfrlemi1.3 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
tfrlemi1.4 (𝜑𝑥 ∈ On)
tfrlemi1.5 (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
Assertion
Ref Expression
tfrlemiex (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))))
Distinct variable groups:   𝑓,𝑔,,𝑢,𝑤,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,,𝑢,𝑤,𝑥,𝑦,𝑧   𝜑,𝑤,𝑦   𝑢,𝐵,𝑤,𝑓,𝑔,,𝑧   𝜑,𝑔,,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑢,𝑓)   𝐵(𝑥,𝑦)

Proof of Theorem tfrlemiex
StepHypRef Expression
1 tfrlemisucfn.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
2 tfrlemisucfn.2 . . . 4 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
3 tfrlemi1.3 . . . 4 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
4 tfrlemi1.4 . . . 4 (𝜑𝑥 ∈ On)
5 tfrlemi1.5 . . . 4 (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
61, 2, 3, 4, 5tfrlemibex 6355 . . 3 (𝜑𝐵 ∈ V)
7 uniexg 4457 . . 3 (𝐵 ∈ V → 𝐵 ∈ V)
86, 7syl 14 . 2 (𝜑 𝐵 ∈ V)
91, 2, 3, 4, 5tfrlemibfn 6354 . . 3 (𝜑 𝐵 Fn 𝑥)
101, 2, 3, 4, 5tfrlemiubacc 6356 . . 3 (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
119, 10jca 306 . 2 (𝜑 → ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
12 fneq1 5323 . . . 4 (𝑓 = 𝐵 → (𝑓 Fn 𝑥 𝐵 Fn 𝑥))
13 fveq1 5533 . . . . . 6 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
14 reseq1 4919 . . . . . . 7 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
1514fveq2d 5538 . . . . . 6 (𝑓 = 𝐵 → (𝐹‘(𝑓𝑢)) = (𝐹‘( 𝐵𝑢)))
1613, 15eqeq12d 2204 . . . . 5 (𝑓 = 𝐵 → ((𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1716ralbidv 2490 . . . 4 (𝑓 = 𝐵 → (∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1812, 17anbi12d 473 . . 3 (𝑓 = 𝐵 → ((𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))) ↔ ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))))
1918spcegv 2840 . 2 ( 𝐵 ∈ V → (( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢)))))
208, 11, 19sylc 62 1 (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980  wal 1362   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wral 2468  wrex 2469  Vcvv 2752  cun 3142  {csn 3607  cop 3610   cuni 3824  Oncon0 4381  cres 4646  Fun wfun 5229   Fn wfn 5230  cfv 5235
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-recs 6331
This theorem is referenced by:  tfrlemi1  6358
  Copyright terms: Public domain W3C validator