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Theorem frecxp 6314
 Description: Subset relationship for the finite recursive function generator. (Contributed by Scott Fenton, 30-Jul-2019.)
Hypotheses
Ref Expression
frecxp.1 F = FRec (G, I)
frecxp.2 G V
Assertion
Ref Expression
frecxp F ( Nn × (ran G ∪ {I}))

Proof of Theorem frecxp
Dummy variables x y z a b c d i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecxp.1 . 2 F = FRec (G, I)
2 eqid 2353 . . . . . 6 G = G
3 freceq12 6311 . . . . . 6 ((G = G i = I) → FRec (G, i) = FRec (G, I))
42, 3mpan 651 . . . . 5 (i = IFRec (G, i) = FRec (G, I))
5 sneq 3744 . . . . . . 7 (i = I → {i} = {I})
65uneq2d 3418 . . . . . 6 (i = I → (ran G ∪ {i}) = (ran G ∪ {I}))
76xpeq2d 4808 . . . . 5 (i = I → ( Nn × (ran G ∪ {i})) = ( Nn × (ran G ∪ {I})))
84, 7sseq12d 3300 . . . 4 (i = I → ( FRec (G, i) ( Nn × (ran G ∪ {i})) ↔ FRec (G, I) ( Nn × (ran G ∪ {I}))))
9 nncex 4396 . . . . . 6 Nn V
10 frecxp.2 . . . . . . . 8 G V
1110rnex 5107 . . . . . . 7 ran G V
12 snex 4111 . . . . . . 7 {i} V
1311, 12unex 4106 . . . . . 6 (ran G ∪ {i}) V
149, 13xpex 5115 . . . . 5 ( Nn × (ran G ∪ {i})) V
15 peano1 4402 . . . . . 6 0c Nn
16 vex 2862 . . . . . . . 8 i V
1716snid 3760 . . . . . . 7 i {i}
18 elun2 3431 . . . . . . 7 (i {i} → i (ran G ∪ {i}))
1917, 18ax-mp 8 . . . . . 6 i (ran G ∪ {i})
20 0cex 4392 . . . . . . . . 9 0c V
2120, 16opex 4588 . . . . . . . 8 0c, i V
2221snss 3838 . . . . . . 7 (0c, i ( Nn × (ran G ∪ {i})) ↔ {0c, i} ( Nn × (ran G ∪ {i})))
23 opelxp 4811 . . . . . . 7 (0c, i ( Nn × (ran G ∪ {i})) ↔ (0c Nn i (ran G ∪ {i})))
2422, 23bitr3i 242 . . . . . 6 ({0c, i} ( Nn × (ran G ∪ {i})) ↔ (0c Nn i (ran G ∪ {i})))
2515, 19, 24mpbir2an 886 . . . . 5 {0c, i} ( Nn × (ran G ∪ {i}))
26 brpprod 5839 . . . . . . . . 9 (y PProd ((x V (x +c 1c)), G)zabcd(y = a, b z = c, d (a(x V (x +c 1c))c bGd)))
27 vex 2862 . . . . . . . . . . . . . . . 16 a V
28 vex 2862 . . . . . . . . . . . . . . . 16 c V
2927, 28brcsuc 6260 . . . . . . . . . . . . . . 15 (a(x V (x +c 1c))cc = (a +c 1c))
30 brelrn 4960 . . . . . . . . . . . . . . . . . . . 20 (bGdd ran G)
31 elun1 3430 . . . . . . . . . . . . . . . . . . . 20 (d ran Gd (ran G ∪ {i}))
3230, 31syl 15 . . . . . . . . . . . . . . . . . . 19 (bGdd (ran G ∪ {i}))
33 peano2 4403 . . . . . . . . . . . . . . . . . . 19 (a Nn → (a +c 1c) Nn )
3432, 33anim12ci 550 . . . . . . . . . . . . . . . . . 18 ((bGd a Nn ) → ((a +c 1c) Nn d (ran G ∪ {i})))
3534adantrr 697 . . . . . . . . . . . . . . . . 17 ((bGd (a Nn b (ran G ∪ {i}))) → ((a +c 1c) Nn d (ran G ∪ {i})))
36 eleq1 2413 . . . . . . . . . . . . . . . . . 18 (c = (a +c 1c) → (c Nn ↔ (a +c 1c) Nn ))
3736anbi1d 685 . . . . . . . . . . . . . . . . 17 (c = (a +c 1c) → ((c Nn d (ran G ∪ {i})) ↔ ((a +c 1c) Nn d (ran G ∪ {i}))))
3835, 37syl5ibr 212 . . . . . . . . . . . . . . . 16 (c = (a +c 1c) → ((bGd (a Nn b (ran G ∪ {i}))) → (c Nn d (ran G ∪ {i}))))
3938exp3a 425 . . . . . . . . . . . . . . 15 (c = (a +c 1c) → (bGd → ((a Nn b (ran G ∪ {i})) → (c Nn d (ran G ∪ {i})))))
4029, 39sylbi 187 . . . . . . . . . . . . . 14 (a(x V (x +c 1c))c → (bGd → ((a Nn b (ran G ∪ {i})) → (c Nn d (ran G ∪ {i})))))
4140imp 418 . . . . . . . . . . . . 13 ((a(x V (x +c 1c))c bGd) → ((a Nn b (ran G ∪ {i})) → (c Nn d (ran G ∪ {i}))))
42 eleq1 2413 . . . . . . . . . . . . . . . 16 (y = a, b → (y ( Nn × (ran G ∪ {i})) ↔ a, b ( Nn × (ran G ∪ {i}))))
43 opelxp 4811 . . . . . . . . . . . . . . . 16 (a, b ( Nn × (ran G ∪ {i})) ↔ (a Nn b (ran G ∪ {i})))
4442, 43syl6bb 252 . . . . . . . . . . . . . . 15 (y = a, b → (y ( Nn × (ran G ∪ {i})) ↔ (a Nn b (ran G ∪ {i}))))
4544adantr 451 . . . . . . . . . . . . . 14 ((y = a, b z = c, d) → (y ( Nn × (ran G ∪ {i})) ↔ (a Nn b (ran G ∪ {i}))))
46 eleq1 2413 . . . . . . . . . . . . . . . 16 (z = c, d → (z ( Nn × (ran G ∪ {i})) ↔ c, d ( Nn × (ran G ∪ {i}))))
47 opelxp 4811 . . . . . . . . . . . . . . . 16 (c, d ( Nn × (ran G ∪ {i})) ↔ (c Nn d (ran G ∪ {i})))
4846, 47syl6bb 252 . . . . . . . . . . . . . . 15 (z = c, d → (z ( Nn × (ran G ∪ {i})) ↔ (c Nn d (ran G ∪ {i}))))
4948adantl 452 . . . . . . . . . . . . . 14 ((y = a, b z = c, d) → (z ( Nn × (ran G ∪ {i})) ↔ (c Nn d (ran G ∪ {i}))))
5045, 49imbi12d 311 . . . . . . . . . . . . 13 ((y = a, b z = c, d) → ((y ( Nn × (ran G ∪ {i})) → z ( Nn × (ran G ∪ {i}))) ↔ ((a Nn b (ran G ∪ {i})) → (c Nn d (ran G ∪ {i})))))
5141, 50syl5ibr 212 . . . . . . . . . . . 12 ((y = a, b z = c, d) → ((a(x V (x +c 1c))c bGd) → (y ( Nn × (ran G ∪ {i})) → z ( Nn × (ran G ∪ {i})))))
52513impia 1148 . . . . . . . . . . 11 ((y = a, b z = c, d (a(x V (x +c 1c))c bGd)) → (y ( Nn × (ran G ∪ {i})) → z ( Nn × (ran G ∪ {i}))))
5352exlimivv 1635 . . . . . . . . . 10 (cd(y = a, b z = c, d (a(x V (x +c 1c))c bGd)) → (y ( Nn × (ran G ∪ {i})) → z ( Nn × (ran G ∪ {i}))))
5453exlimivv 1635 . . . . . . . . 9 (abcd(y = a, b z = c, d (a(x V (x +c 1c))c bGd)) → (y ( Nn × (ran G ∪ {i})) → z ( Nn × (ran G ∪ {i}))))
5526, 54sylbi 187 . . . . . . . 8 (y PProd ((x V (x +c 1c)), G)z → (y ( Nn × (ran G ∪ {i})) → z ( Nn × (ran G ∪ {i}))))
5655impcom 419 . . . . . . 7 ((y ( Nn × (ran G ∪ {i})) y PProd ((x V (x +c 1c)), G)z) → z ( Nn × (ran G ∪ {i})))
5756ax-gen 1546 . . . . . 6 z((y ( Nn × (ran G ∪ {i})) y PProd ((x V (x +c 1c)), G)z) → z ( Nn × (ran G ∪ {i})))
5857rgenw 2681 . . . . 5 y FRec (G, i)z((y ( Nn × (ran G ∪ {i})) y PProd ((x V (x +c 1c)), G)z) → z ( Nn × (ran G ∪ {i})))
59 snex 4111 . . . . . 6 {0c, i} V
60 csucex 6259 . . . . . . 7 (x V (x +c 1c)) V
6160, 10pprodex 5838 . . . . . 6 PProd ((x V (x +c 1c)), G) V
62 df-frec 6310 . . . . . 6 FRec (G, i) = Clos1 ({0c, i}, PProd ((x V (x +c 1c)), G))
6359, 61, 62clos1induct 5880 . . . . 5 ((( Nn × (ran G ∪ {i})) V {0c, i} ( Nn × (ran G ∪ {i})) y FRec (G, i)z((y ( Nn × (ran G ∪ {i})) y PProd ((x V (x +c 1c)), G)z) → z ( Nn × (ran G ∪ {i})))) → FRec (G, i) ( Nn × (ran G ∪ {i})))
6414, 25, 58, 63mp3an 1277 . . . 4 FRec (G, i) ( Nn × (ran G ∪ {i}))
658, 64vtoclg 2914 . . 3 (I V → FRec (G, I) ( Nn × (ran G ∪ {I})))
66 df-frec 6310 . . . 4 FRec (G, I) = Clos1 ({0c, I}, PProd ((x V (x +c 1c)), G))
67 opexb 4603 . . . . . . . . . 10 (0c, I V ↔ (0c V I V))
6867simprbi 450 . . . . . . . . 9 (0c, I V → I V)
6968con3i 127 . . . . . . . 8 I V → ¬ 0c, I V)
70 snprc 3788 . . . . . . . 8 0c, I V ↔ {0c, I} = )
7169, 70sylib 188 . . . . . . 7 I V → {0c, I} = )
72 clos1eq1 5874 . . . . . . 7 ({0c, I} = Clos1 ({0c, I}, PProd ((x V (x +c 1c)), G)) = Clos1 (, PProd ((x V (x +c 1c)), G)))
7371, 72syl 15 . . . . . 6 I V → Clos1 ({0c, I}, PProd ((x V (x +c 1c)), G)) = Clos1 (, PProd ((x V (x +c 1c)), G)))
74 eqid 2353 . . . . . . 7 Clos1 (, PProd ((x V (x +c 1c)), G)) = Clos1 (, PProd ((x V (x +c 1c)), G))
7561, 74clos10 5887 . . . . . 6 Clos1 (, PProd ((x V (x +c 1c)), G)) =
7673, 75syl6eq 2401 . . . . 5 I V → Clos1 ({0c, I}, PProd ((x V (x +c 1c)), G)) = )
77 0ss 3579 . . . . 5 ( Nn × (ran G ∪ {I}))
7876, 77syl6eqss 3321 . . . 4 I V → Clos1 ({0c, I}, PProd ((x V (x +c 1c)), G)) ( Nn × (ran G ∪ {I})))
7966, 78syl5eqss 3315 . . 3 I V → FRec (G, I) ( Nn × (ran G ∪ {I})))
8065, 79pm2.61i 156 . 2 FRec (G, I) ( Nn × (ran G ∪ {I}))
811, 80eqsstri 3301 1 F ( Nn × (ran G ∪ {I}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ∪ cun 3207   ⊆ wss 3257  ∅c0 3550  {csn 3737  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375  ⟨cop 4561   class class class wbr 4639   × cxp 4770  ran crn 4773   ↦ cmpt 5651   PProd cpprod 5737   Clos1 cclos1 5872   FRec cfrec 6309 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-pprod 5738  df-fix 5740  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-clos1 5873  df-frec 6310 This theorem is referenced by:  frecxpg  6315
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