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Theorem frecxpg 6315
 Description: Subset relationship for the finite recursive function generator. (Contributed by Scott Fenton, 31-Jul-2019.)
Hypothesis
Ref Expression
frecxpg.1 F = FRec (G, I)
Assertion
Ref Expression
frecxpg (G VF ( Nn × (ran G ∪ {I})))

Proof of Theorem frecxpg
Dummy variable g is distinct from all other variables.
StepHypRef Expression
1 frecxpg.1 . 2 F = FRec (G, I)
2 eqid 2353 . . . . 5 I = I
3 freceq12 6311 . . . . 5 ((g = G I = I) → FRec (g, I) = FRec (G, I))
42, 3mpan2 652 . . . 4 (g = GFRec (g, I) = FRec (G, I))
5 rneq 4956 . . . . . 6 (g = G → ran g = ran G)
65uneq1d 3417 . . . . 5 (g = G → (ran g ∪ {I}) = (ran G ∪ {I}))
76xpeq2d 4808 . . . 4 (g = G → ( Nn × (ran g ∪ {I})) = ( Nn × (ran G ∪ {I})))
84, 7sseq12d 3300 . . 3 (g = G → ( FRec (g, I) ( Nn × (ran g ∪ {I})) ↔ FRec (G, I) ( Nn × (ran G ∪ {I}))))
9 eqid 2353 . . . 4 FRec (g, I) = FRec (g, I)
10 vex 2862 . . . 4 g V
119, 10frecxp 6314 . . 3 FRec (g, I) ( Nn × (ran g ∪ {I}))
128, 11vtoclg 2914 . 2 (G VFRec (G, I) ( Nn × (ran G ∪ {I})))
131, 12syl5eqss 3315 1 (G VF ( Nn × (ran G ∪ {I})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710   ∪ cun 3207   ⊆ wss 3257  {csn 3737   Nn cnnc 4373   × cxp 4770  ran crn 4773   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:  dmfrec  6316  frecsuc  6322
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