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Theorem tfrlem7 8422
Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem7 Fun recs(𝐹)
Distinct variable group:   𝑥,𝑓,𝑦,𝐹
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem7
Dummy variables 𝑔 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem6 8421 . 2 Rel recs(𝐹)
31recsfval 8420 . . . . . . . . 9 recs(𝐹) = 𝐴
43eleq2i 2831 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐴)
5 eluni 4915 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ 𝐴 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴))
64, 5bitri 275 . . . . . . 7 (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴))
73eleq2i 2831 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐴)
8 eluni 4915 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ 𝐴 ↔ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴))
97, 8bitri 275 . . . . . . 7 (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴))
106, 9anbi12i 628 . . . . . 6 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴)))
11 exdistrv 1953 . . . . . 6 (∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴)))
1210, 11bitr4i 278 . . . . 5 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ ∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)))
13 df-br 5149 . . . . . . . . 9 (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
14 df-br 5149 . . . . . . . . 9 (𝑥𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ )
1513, 14anbi12i 628 . . . . . . . 8 ((𝑥𝑔𝑢𝑥𝑣) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ))
161tfrlem5 8419 . . . . . . . . 9 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
1716impcom 407 . . . . . . . 8 (((𝑥𝑔𝑢𝑥𝑣) ∧ (𝑔𝐴𝐴)) → 𝑢 = 𝑣)
1815, 17sylanbr 582 . . . . . . 7 (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ) ∧ (𝑔𝐴𝐴)) → 𝑢 = 𝑣)
1918an4s 660 . . . . . 6 (((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) → 𝑢 = 𝑣)
2019exlimivv 1930 . . . . 5 (∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) → 𝑢 = 𝑣)
2112, 20sylbi 217 . . . 4 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
2221ax-gen 1792 . . 3 𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
2322gen2 1793 . 2 𝑥𝑢𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
24 dffun4 6579 . 2 (Fun recs(𝐹) ↔ (Rel recs(𝐹) ∧ ∀𝑥𝑢𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)))
252, 23, 24mpbir2an 711 1 Fun recs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  cop 4637   cuni 4912   class class class wbr 5148  cres 5691  Rel wrel 5694  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  cfv 6563  recscrecs 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410
This theorem is referenced by:  tfrlem9  8424  tfrlem9a  8425  tfrlem10  8426  tfrlem14  8430  tfrlem16  8432  tfr1a  8433  tfr1  8436
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