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Theorem tfrlem5 5871
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 2554 . . 3 g V
31, 2tfrlem3a 5866 . 2 (g Az On (g Fn z 𝑎 z (g𝑎) = (𝐹‘(g𝑎))))
4 vex 2554 . . 3 V
51, 4tfrlem3a 5866 . 2 ( Aw On ( Fn w 𝑎 w (𝑎) = (𝐹‘(𝑎))))
6 reeanv 2473 . . 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 970 . . . . . . . . . 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 931 . . . . . . . . . 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 4944 . . . . . . . . . 10 ((g Fn z xgu) → x z)
107, 8, 9syl2anc 391 . . . . . . . . 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 972 . . . . . . . . . 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 932 . . . . . . . . . 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 4944 . . . . . . . . . 10 (( Fn w xv) → x w)
1411, 12, 13syl2anc 391 . . . . . . . . 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 3120 . . . . . . . . 9 (x (zw) ↔ (x z x w))
1610, 14, 15sylanbrc 394 . . . . . . . 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 4089 . . . . . . . . . 10 ((z On w On) → (zw) On)
18173ad2ant1 924 . . . . . . . . 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 4939 . . . . . . . . . . 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 3151 . . . . . . . . . . 11 (zw) ⊆ z
22 fndm 4941 . . . . . . . . . . . 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 2988 . . . . . . . . . 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 4939 . . . . . . . . . . 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 3152 . . . . . . . . . . 11 (zw) ⊆ w
29 fndm 4941 . . . . . . . . . . . 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 2988 . . . . . . . . . 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 971 . . . . . . . . . 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 2998 . . . . . . . . . 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 973 . . . . . . . . . 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 2998 . . . . . . . . . 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 5864 . . . . . . . 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 5121 . . . . . . . . . 10 (𝑎 = x → (g𝑎) = (gx))
41 fveq2 5121 . . . . . . . . . 10 (𝑎 = x → (𝑎) = (x))
4240, 41eqeq12d 2051 . . . . . . . . 9 (𝑎 = x → ((g𝑎) = (𝑎) ↔ (gx) = (x)))
4342rspcv 2646 . . . . . . . 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 5155 . . . . . . . 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 5155 . . . . . . . 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 2077 . . . . . 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 1102 . . . . 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 2427 . . . 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 2421 . . 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 884   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  cin 2910  wss 2911   class class class wbr 3755  Oncon0 4066  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  tfrlem7  5874  tfrexlem  5889
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