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Theorem frrlem7 8279
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 8277 . . . . . 6 𝐹 = 𝐵
43dmeqi 5903 . . . . 5 dom 𝐹 = dom 𝐵
5 dmuni 5913 . . . . 5 dom 𝐵 = 𝑔𝐵 dom 𝑔
64, 5eqtri 2758 . . . 4 dom 𝐹 = 𝑔𝐵 dom 𝑔
76sseq1i 4009 . . 3 (dom 𝐹𝐴 𝑔𝐵 dom 𝑔𝐴)
8 iunss 5047 . . 3 ( 𝑔𝐵 dom 𝑔𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
97, 8bitri 274 . 2 (dom 𝐹𝐴 ↔ ∀𝑔𝐵 dom 𝑔𝐴)
101frrlem3 8275 . 2 (𝑔𝐵 → dom 𝑔𝐴)
119, 10mprgbir 3066 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 394  w3a 1085   = wceq 1539  wex 1779  {cab 2707  wral 3059  wss 3947   cuni 4907   ciun 4996  dom cdm 5675  cres 5677  Predcpred 6298   Fn wfn 6537  cfv 6542  (class class class)co 7411  frecscfrecs 8267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ov 7414  df-frecs 8268
This theorem is referenced by:  frrlem14  8286  frrdmss  8294
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