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Theorem tfrlem7 7246
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 7245 . 2 Rel recs(𝐹)
31recsfval 7244 . . . . . . . . 9 recs(𝐹) = 𝐴
43eleq2i 2584 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐴)
5 eluni 4273 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ 𝐴 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴))
64, 5bitri 262 . . . . . . 7 (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴))
73eleq2i 2584 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐴)
8 eluni 4273 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ 𝐴 ↔ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴))
97, 8bitri 262 . . . . . . 7 (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴))
106, 9anbi12i 728 . . . . . 6 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴)))
11 eeanv 2125 . . . . . 6 (∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴)))
1210, 11bitr4i 265 . . . . 5 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ ∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)))
13 df-br 4482 . . . . . . . . 9 (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
14 df-br 4482 . . . . . . . . 9 (𝑥𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ )
1513, 14anbi12i 728 . . . . . . . 8 ((𝑥𝑔𝑢𝑥𝑣) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ))
161tfrlem5 7243 . . . . . . . . 9 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
1716impcom 444 . . . . . . . 8 (((𝑥𝑔𝑢𝑥𝑣) ∧ (𝑔𝐴𝐴)) → 𝑢 = 𝑣)
1815, 17sylanbr 488 . . . . . . 7 (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ) ∧ (𝑔𝐴𝐴)) → 𝑢 = 𝑣)
1918an4s 864 . . . . . 6 (((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) → 𝑢 = 𝑣)
2019exlimivv 1813 . . . . 5 (∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) → 𝑢 = 𝑣)
2112, 20sylbi 205 . . . 4 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
2221ax-gen 1700 . . 3 𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
2322gen2 1701 . 2 𝑥𝑢𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
24 dffun4 5706 . 2 (Fun recs(𝐹) ↔ (Rel recs(𝐹) ∧ ∀𝑥𝑢𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)))
252, 23, 24mpbir2an 956 1 Fun recs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1938  {cab 2500  wral 2800  wrex 2801  cop 4034   cuni 4270   class class class wbr 4481  cres 4934  Rel wrel 4937  Oncon0 5530  Fun wfun 5688   Fn wfn 5689  cfv 5694  recscrecs 7234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6728
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-iota 5658  df-fun 5696  df-fn 5697  df-fv 5702  df-wrecs 7174  df-recs 7235
This theorem is referenced by:  tfrlem9  7248  tfrlem9a  7249  tfrlem10  7250  tfrlem14  7254  tfrlem16  7256  tfr1a  7257  tfr1  7260
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