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Theorem tfr1onlemex 6338
Description: Lemma for tfr1on 6341. (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 6336 . . 3 (𝜑𝐵 ∈ V)
11 uniexg 4433 . . 3 (𝐵 ∈ V → 𝐵 ∈ V)
1210, 11syl 14 . 2 (𝜑 𝐵 ∈ V)
131, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembfn 6335 . . 3 (𝜑 𝐵 Fn 𝐷)
141, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlemubacc 6337 . . 3 (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
1513, 14jca 306 . 2 (𝜑 → ( 𝐵 Fn 𝐷 ∧ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))))
16 fneq1 5296 . . . 4 (𝑓 = 𝐵 → (𝑓 Fn 𝐷 𝐵 Fn 𝐷))
17 fveq1 5506 . . . . . 6 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
18 reseq1 4894 . . . . . . 7 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
1918fveq2d 5511 . . . . . 6 (𝑓 = 𝐵 → (𝐺‘(𝑓𝑢)) = (𝐺‘( 𝐵𝑢)))
2017, 19eqeq12d 2190 . . . . 5 (𝑓 = 𝐵 → ((𝑓𝑢) = (𝐺‘(𝑓𝑢)) ↔ ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))))
2120ralbidv 2475 . . . 4 (𝑓 = 𝐵 → (∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢)) ↔ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))))
2216, 21anbi12d 473 . . 3 (𝑓 = 𝐵 → ((𝑓 Fn 𝐷 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢))) ↔ ( 𝐵 Fn 𝐷 ∧ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))))
2322spcegv 2823 . 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 104  w3a 978   = wceq 1353  wex 1490  wcel 2146  {cab 2161  wral 2453  wrex 2454  Vcvv 2735  cun 3125  {csn 3589  cop 3592   cuni 3805  Ord word 4356  suc csuc 4359  cres 4622  Fun wfun 5202   Fn wfn 5203  cfv 5208  recscrecs 6295
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-recs 6296
This theorem is referenced by:  tfr1onlemaccex  6339
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