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Theorem frrlem7 8318
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 8316 . . . . . 6 𝐹 = 𝐵
43dmeqi 5914 . . . . 5 dom 𝐹 = dom 𝐵
5 dmuni 5924 . . . . 5 dom 𝐵 = 𝑔𝐵 dom 𝑔
64, 5eqtri 2764 . . . 4 dom 𝐹 = 𝑔𝐵 dom 𝑔
76sseq1i 4011 . . 3 (dom 𝐹𝐴 𝑔𝐵 dom 𝑔𝐴)
8 iunss 5044 . . 3 ( 𝑔𝐵 dom 𝑔𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
97, 8bitri 275 . 2 (dom 𝐹𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
101frrlem3 8314 . 2 (𝑔𝐵 → dom 𝑔𝐴)
119, 10mprgbir 3067 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1086   = wceq 1539  wex 1778  {cab 2713  wral 3060  wss 3950   cuni 4906   ciun 4990  dom cdm 5684  cres 5686  Predcpred 6319   Fn wfn 6555  cfv 6560  (class class class)co 7432  frecscfrecs 8306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ov 7435  df-frecs 8307
This theorem is referenced by:  frrlem14  8325  frrdmss  8333
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