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Theorem frrlem7 8277
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 8275 . . . . . 6 𝐹 = 𝐵
43dmeqi 5905 . . . . 5 dom 𝐹 = dom 𝐵
5 dmuni 5915 . . . . 5 dom 𝐵 = 𝑔𝐵 dom 𝑔
64, 5eqtri 2761 . . . 4 dom 𝐹 = 𝑔𝐵 dom 𝑔
76sseq1i 4011 . . 3 (dom 𝐹𝐴 𝑔𝐵 dom 𝑔𝐴)
8 iunss 5049 . . 3 ( 𝑔𝐵 dom 𝑔𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
97, 8bitri 275 . 2 (dom 𝐹𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
101frrlem3 8273 . 2 (𝑔𝐵 → dom 𝑔𝐴)
119, 10mprgbir 3069 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 397  w3a 1088   = wceq 1542  wex 1782  {cab 2710  wral 3062  wss 3949   cuni 4909   ciun 4998  dom cdm 5677  cres 5679  Predcpred 6300   Fn wfn 6539  cfv 6544  (class class class)co 7409  frecscfrecs 8265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-ov 7412  df-frecs 8266
This theorem is referenced by:  frrlem14  8284  frrdmss  8292
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