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Theorem tfrlem5 5852
Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (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
tfrlem5 ((g A A) → ((xgu xv) → u = v))
Distinct variable groups:   f,g,x,y,,u,v,𝐹   A,g,
Allowed substitution hints:   A(x,y,v,u,f)

Proof of Theorem tfrlem5
Dummy variables z 𝑎 w 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)))}
2 vex 2538 . . 3 g V
31, 2tfrlem3a 5847 . 2 (g Az On (g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))))
4 vex 2538 . . 3 V
51, 4tfrlem3a 5847 . 2 ( Aw On ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎))))
6 reeanv 2457 . . 3 (z On w On ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) ↔ (z On (g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) w On ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))))
7 simp2ll 959 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → g Fn z)
8 simp3l 920 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → xgu)
9 fnbr 4927 . . . . . . . . . 10 ((g Fn z xgu) → x z)
107, 8, 9syl2anc 393 . . . . . . . . 9 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → x z)
11 simp2rl 961 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → Fn w)
12 simp3r 921 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → xv)
13 fnbr 4927 . . . . . . . . . 10 (( Fn w xv) → x w)
1411, 12, 13syl2anc 393 . . . . . . . . 9 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → x w)
15 elin 3103 . . . . . . . . 9 (x (zw) ↔ (x z x w))
1610, 14, 15sylanbrc 396 . . . . . . . 8 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → x (zw))
17 onin 4072 . . . . . . . . . 10 ((z On w On) → (zw) On)
18173ad2ant1 913 . . . . . . . . 9 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → (zw) On)
19 fnfun 4922 . . . . . . . . . . 11 (g Fn z → Fun g)
207, 19syl 14 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → Fun g)
21 inss1 3134 . . . . . . . . . . 11 (zw) ⊆ z
22 fndm 4924 . . . . . . . . . . . 12 (g Fn z → dom g = z)
237, 22syl 14 . . . . . . . . . . 11 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → dom g = z)
2421, 23syl5sseqr 2971 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → (zw) ⊆ dom g)
2520, 24jca 290 . . . . . . . . 9 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → (Fun g (zw) ⊆ dom g))
26 fnfun 4922 . . . . . . . . . . 11 ( Fn w → Fun )
2711, 26syl 14 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → Fun )
28 inss2 3135 . . . . . . . . . . 11 (zw) ⊆ w
29 fndm 4924 . . . . . . . . . . . 12 ( Fn w → dom = w)
3011, 29syl 14 . . . . . . . . . . 11 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → dom = w)
3128, 30syl5sseqr 2971 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → (zw) ⊆ dom )
3227, 31jca 290 . . . . . . . . 9 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → (Fun (zw) ⊆ dom ))
33 simp2lr 960 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → 𝑎 z (g𝑎) = (𝐹‘(g𝑎)))
34 ssralv 2981 . . . . . . . . . 10 ((zw) ⊆ z → (𝑎 z (g𝑎) = (𝐹‘(g𝑎)) → 𝑎 (zw)(g𝑎) = (𝐹‘(g𝑎))))
3521, 33, 34mpsyl 59 . . . . . . . . 9 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → 𝑎 (zw)(g𝑎) = (𝐹‘(g𝑎)))
36 simp2rr 962 . . . . . . . . . 10 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → 𝑎 w (𝑎) = (𝐹‘(𝑎)))
37 ssralv 2981 . . . . . . . . . 10 ((zw) ⊆ w → (𝑎 w (𝑎) = (𝐹‘(𝑎)) → 𝑎 (zw)(𝑎) = (𝐹‘(𝑎))))
3828, 36, 37mpsyl 59 . . . . . . . . 9 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → 𝑎 (zw)(𝑎) = (𝐹‘(𝑎)))
3918, 25, 32, 35, 38tfrlem1 5845 . . . . . . . 8 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → 𝑎 (zw)(g𝑎) = (𝑎))
40 fveq2 5103 . . . . . . . . . 10 (𝑎 = x → (g𝑎) = (gx))
41 fveq2 5103 . . . . . . . . . 10 (𝑎 = x → (𝑎) = (x))
4240, 41eqeq12d 2036 . . . . . . . . 9 (𝑎 = x → ((g𝑎) = (𝑎) ↔ (gx) = (x)))
4342rspcv 2629 . . . . . . . 8 (x (zw) → (𝑎 (zw)(g𝑎) = (𝑎) → (gx) = (x)))
4416, 39, 43sylc 56 . . . . . . 7 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → (gx) = (x))
45 funbrfv 5137 . . . . . . . 8 (Fun g → (xgu → (gx) = u))
4620, 8, 45sylc 56 . . . . . . 7 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → (gx) = u)
47 funbrfv 5137 . . . . . . . 8 (Fun → (xv → (x) = v))
4827, 12, 47sylc 56 . . . . . . 7 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → (x) = v)
4944, 46, 483eqtr3d 2062 . . . . . 6 (((z On w On) ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) (xgu xv)) → u = v)
50493exp 1089 . . . . 5 ((z On w On) → (((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) → ((xgu xv) → u = v)))
5150rexlimdva 2411 . . . 4 (z On → (w On ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) → ((xgu xv) → u = v)))
5251rexlimiv 2405 . . 3 (z On w On ((g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) → ((xgu xv) → u = v))
536, 52sylbir 125 . 2 ((z On (g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))) w On ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎)))) → ((xgu xv) → u = v))
543, 5, 53syl2anb 275 1 ((g A A) → ((xgu xv) → u = v))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873   = wceq 1228   wcel 1374  {cab 2008  wral 2284  wrex 2285  cin 2893  wss 2894   class class class wbr 3738  Oncon0 4049  dom cdm 4272  cres 4274  Fun wfun 4823   Fn wfn 4824  cfv 4829
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-res 4284  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  tfrlem7  5855  tfrexlem  5870
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