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Theorem frrlem7 8316
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 8314 . . . . . 6 𝐹 = 𝐵
43dmeqi 5918 . . . . 5 dom 𝐹 = dom 𝐵
5 dmuni 5928 . . . . 5 dom 𝐵 = 𝑔𝐵 dom 𝑔
64, 5eqtri 2763 . . . 4 dom 𝐹 = 𝑔𝐵 dom 𝑔
76sseq1i 4024 . . 3 (dom 𝐹𝐴 𝑔𝐵 dom 𝑔𝐴)
8 iunss 5050 . . 3 ( 𝑔𝐵 dom 𝑔𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
97, 8bitri 275 . 2 (dom 𝐹𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
101frrlem3 8312 . 2 (𝑔𝐵 → dom 𝑔𝐴)
119, 10mprgbir 3066 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1086   = wceq 1537  wex 1776  {cab 2712  wral 3059  wss 3963   cuni 4912   ciun 4996  dom cdm 5689  cres 5691  Predcpred 6322   Fn wfn 6558  cfv 6563  (class class class)co 7431  frecscfrecs 8304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-frecs 8305
This theorem is referenced by:  frrlem14  8323  frrdmss  8331
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