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Theorem tfr1onlemex 6212
Description: Lemma for tfr1on 6215. (Contributed by Jim Kingdon, 16-Mar-2022.)
Hypotheses
Ref Expression
tfr1on.f 𝐹 = recs(𝐺)
tfr1on.g (𝜑 → Fun 𝐺)
tfr1on.x (𝜑 → Ord 𝑋)
tfr1on.ex ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
tfr1onlemsucfn.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
tfr1onlembacc.3 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
tfr1onlembacc.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfr1onlembacc.4 (𝜑𝐷𝑋)
tfr1onlembacc.5 (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
Assertion
Ref Expression
tfr1onlemex (𝜑 → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢))))
Distinct variable groups:   𝐴,𝑓,𝑔,,𝑥,𝑧   𝐷,𝑓,𝑔,𝑥   𝑓,𝐺,𝑥,𝑦   𝑓,𝑋,𝑥   𝜑,𝑓,𝑔,,𝑥,𝑧   𝑦,𝑔,𝑧   𝐵,𝑓,𝑔,,𝑤,𝑧   𝑢,𝐵,𝑓,𝑤   𝐷,,𝑤,𝑧,𝑥   𝑢,𝐷   ,𝐺,𝑧,𝑦   𝑢,𝐺,𝑤   𝑔,𝑋,𝑧   𝜑,𝑤   𝑦,𝑤
Allowed substitution hints:   𝜑(𝑦,𝑢)   𝐴(𝑦,𝑤,𝑢)   𝐵(𝑥,𝑦)   𝐷(𝑦)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑢,𝑓,𝑔,)   𝐺(𝑔)   𝑋(𝑦,𝑤,𝑢,)

Proof of Theorem tfr1onlemex
StepHypRef Expression
1 tfr1on.f . . . 4 𝐹 = recs(𝐺)
2 tfr1on.g . . . 4 (𝜑 → Fun 𝐺)
3 tfr1on.x . . . 4 (𝜑 → Ord 𝑋)
4 tfr1on.ex . . . 4 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
5 tfr1onlemsucfn.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
6 tfr1onlembacc.3 . . . 4 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
7 tfr1onlembacc.u . . . 4 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
8 tfr1onlembacc.4 . . . 4 (𝜑𝐷𝑋)
9 tfr1onlembacc.5 . . . 4 (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
101, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembex 6210 . . 3 (𝜑𝐵 ∈ V)
11 uniexg 4331 . . 3 (𝐵 ∈ V → 𝐵 ∈ V)
1210, 11syl 14 . 2 (𝜑 𝐵 ∈ V)
131, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembfn 6209 . . 3 (𝜑 𝐵 Fn 𝐷)
141, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlemubacc 6211 . . 3 (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
1513, 14jca 304 . 2 (𝜑 → ( 𝐵 Fn 𝐷 ∧ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))))
16 fneq1 5181 . . . 4 (𝑓 = 𝐵 → (𝑓 Fn 𝐷 𝐵 Fn 𝐷))
17 fveq1 5388 . . . . . 6 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
18 reseq1 4783 . . . . . . 7 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
1918fveq2d 5393 . . . . . 6 (𝑓 = 𝐵 → (𝐺‘(𝑓𝑢)) = (𝐺‘( 𝐵𝑢)))
2017, 19eqeq12d 2132 . . . . 5 (𝑓 = 𝐵 → ((𝑓𝑢) = (𝐺‘(𝑓𝑢)) ↔ ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))))
2120ralbidv 2414 . . . 4 (𝑓 = 𝐵 → (∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢)) ↔ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))))
2216, 21anbi12d 464 . . 3 (𝑓 = 𝐵 → ((𝑓 Fn 𝐷 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢))) ↔ ( 𝐵 Fn 𝐷 ∧ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))))
2322spcegv 2748 . 2 ( 𝐵 ∈ V → (( 𝐵 Fn 𝐷 ∧ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))) → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢)))))
2412, 15, 23sylc 62 1 (𝜑 → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wral 2393  wrex 2394  Vcvv 2660  cun 3039  {csn 3497  cop 3500   cuni 3706  Ord word 4254  suc csuc 4257  cres 4511  Fun wfun 5087   Fn wfn 5088  cfv 5093  recscrecs 6169
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:  tfr1onlemaccex  6213
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