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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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