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

Theorem tfrlemiex 6196
Description: Lemma for tfrlemi1 6197. (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 6194 . . 3 (𝜑𝐵 ∈ V)
7 uniexg 4331 . . 3 (𝐵 ∈ V → 𝐵 ∈ V)
86, 7syl 14 . 2 (𝜑 𝐵 ∈ V)
91, 2, 3, 4, 5tfrlemibfn 6193 . . 3 (𝜑 𝐵 Fn 𝑥)
101, 2, 3, 4, 5tfrlemiubacc 6195 . . 3 (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
119, 10jca 304 . 2 (𝜑 → ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
12 fneq1 5181 . . . 4 (𝑓 = 𝐵 → (𝑓 Fn 𝑥 𝐵 Fn 𝑥))
13 fveq1 5388 . . . . . 6 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
14 reseq1 4783 . . . . . . 7 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
1514fveq2d 5393 . . . . . 6 (𝑓 = 𝐵 → (𝐹‘(𝑓𝑢)) = (𝐹‘( 𝐵𝑢)))
1613, 15eqeq12d 2132 . . . . 5 (𝑓 = 𝐵 → ((𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1716ralbidv 2414 . . . 4 (𝑓 = 𝐵 → (∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1812, 17anbi12d 464 . . 3 (𝑓 = 𝐵 → ((𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))) ↔ ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))))
1918spcegv 2748 . 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 103  w3a 947  wal 1314   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wral 2393  wrex 2394  Vcvv 2660  cun 3039  {csn 3497  cop 3500   cuni 3706  Oncon0 4255  cres 4511  Fun wfun 5087   Fn wfn 5088  cfv 5093
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170
This theorem is referenced by:  tfrlemi1  6197
  Copyright terms: Public domain W3C validator