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Theorem tfrlem7 5851
 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 = {fx On (f Fn x y x (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 = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
21tfrlem6 5850 . 2 Rel recs(𝐹)
31recsfval 5849 . . . . . . . . 9 recs(𝐹) = A
43eleq2i 2082 . . . . . . . 8 (⟨x, u recs(𝐹) ↔ ⟨x, u A)
5 eluni 3553 . . . . . . . 8 (⟨x, u Ag(⟨x, u g g A))
64, 5bitri 173 . . . . . . 7 (⟨x, u recs(𝐹) ↔ g(⟨x, u g g A))
73eleq2i 2082 . . . . . . . 8 (⟨x, v recs(𝐹) ↔ ⟨x, v A)
8 eluni 3553 . . . . . . . 8 (⟨x, v A(⟨x, v A))
97, 8bitri 173 . . . . . . 7 (⟨x, v recs(𝐹) ↔ (⟨x, v A))
106, 9anbi12i 436 . . . . . 6 ((⟨x, u recs(𝐹) x, v recs(𝐹)) ↔ (g(⟨x, u g g A) (⟨x, v A)))
11 eeanv 1785 . . . . . 6 (g((⟨x, u g g A) (⟨x, v A)) ↔ (g(⟨x, u g g A) (⟨x, v A)))
1210, 11bitr4i 176 . . . . 5 ((⟨x, u recs(𝐹) x, v recs(𝐹)) ↔ g((⟨x, u g g A) (⟨x, v A)))
13 df-br 3735 . . . . . . . . 9 (xgu ↔ ⟨x, u g)
14 df-br 3735 . . . . . . . . 9 (xv ↔ ⟨x, v )
1513, 14anbi12i 436 . . . . . . . 8 ((xgu xv) ↔ (⟨x, u g x, v ))
161tfrlem5 5848 . . . . . . . . 9 ((g A A) → ((xgu xv) → u = v))
1716impcom 116 . . . . . . . 8 (((xgu xv) (g A A)) → u = v)
1815, 17sylanbr 269 . . . . . . 7 (((⟨x, u g x, v ) (g A A)) → u = v)
1918an4s 509 . . . . . 6 (((⟨x, u g g A) (⟨x, v A)) → u = v)
2019exlimivv 1754 . . . . 5 (g((⟨x, u g g A) (⟨x, v A)) → u = v)
2112, 20sylbi 114 . . . 4 ((⟨x, u recs(𝐹) x, v recs(𝐹)) → u = v)
2221ax-gen 1314 . . 3 v((⟨x, u recs(𝐹) x, v recs(𝐹)) → u = v)
2322gen2 1315 . 2 xuv((⟨x, u recs(𝐹) x, v recs(𝐹)) → u = v)
24 dffun4 4836 . 2 (Fun recs(𝐹) ↔ (Rel recs(𝐹) xuv((⟨x, u recs(𝐹) x, v recs(𝐹)) → u = v)))
252, 23, 24mpbir2an 835 1 Fun recs(𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1224   = wceq 1226  ∃wex 1358   ∈ wcel 1370  {cab 2004  ∀wral 2280  ∃wrex 2281  ⟨cop 3349  ∪ cuni 3550   class class class wbr 3734  Oncon0 4045   ↾ cres 4270  Rel wrel 4273  Fun wfun 4819   Fn wfn 4820  ‘cfv 4825  recscrecs 5837 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-setind 4200 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-res 4280  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833  df-recs 5838 This theorem is referenced by:  tfrlem9  5853  tfrlemibfn  5859  tfrlemiubacc  5861  tfri1d  5867  tfri1  5869  rdgfun  5877
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