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Theorem frrlem7 8307
Description: Lemma for well-founded recursion. The well-founded recursion generator's domain is a subclass of 𝐴. (Contributed by Scott Fenton, 27-Aug-2022.)
Hypotheses
Ref Expression
frrlem5.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem5.2 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
frrlem7 dom 𝐹𝐴
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑓)   𝐹(𝑥,𝑦,𝑓)

Proof of Theorem frrlem7
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 frrlem5.1 . . . . . . 7 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2 frrlem5.2 . . . . . . 7 𝐹 = frecs(𝑅, 𝐴, 𝐺)
31, 2frrlem5 8305 . . . . . 6 𝐹 = 𝐵
43dmeqi 5911 . . . . 5 dom 𝐹 = dom 𝐵
5 dmuni 5921 . . . . 5 dom 𝐵 = 𝑔𝐵 dom 𝑔
64, 5eqtri 2754 . . . 4 dom 𝐹 = 𝑔𝐵 dom 𝑔
76sseq1i 4008 . . 3 (dom 𝐹𝐴 𝑔𝐵 dom 𝑔𝐴)
8 iunss 5053 . . 3 ( 𝑔𝐵 dom 𝑔𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
97, 8bitri 274 . 2 (dom 𝐹𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
101frrlem3 8303 . 2 (𝑔𝐵 → dom 𝑔𝐴)
119, 10mprgbir 3058 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 394  w3a 1084   = wceq 1534  wex 1774  {cab 2703  wral 3051  wss 3947   cuni 4913   ciun 5001  dom cdm 5682  cres 5684  Predcpred 6311   Fn wfn 6549  cfv 6554  (class class class)co 7424  frecscfrecs 8295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-iota 6506  df-fun 6556  df-fn 6557  df-fv 6562  df-ov 7427  df-frecs 8296
This theorem is referenced by:  frrlem14  8314  frrdmss  8322
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