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Theorem frrlem3 33182
Description: Lemma for founded recursion. An acceptable function's domain is a subset of 𝐴. (Contributed by Paul Chapman, 21-Apr-2012.)
Hypothesis
Ref Expression
frrlem1.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Assertion
Ref Expression
frrlem3 (𝑔𝐵 → dom 𝑔𝐴)
Distinct variable groups:   𝐴,𝑓,𝑔,𝑥,𝑦   𝑓,𝐺,𝑔,𝑥,𝑦   𝑅,𝑓,𝑔,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem frrlem3
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frrlem1.1 . . . 4 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
21frrlem1 33180 . . 3 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
32abeq2i 2951 . 2 (𝑔𝐵 ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
4 fndm 6443 . . . . . . 7 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
54sseq1d 3984 . . . . . 6 (𝑔 Fn 𝑧 → (dom 𝑔𝐴𝑧𝐴))
65biimpar 481 . . . . 5 ((𝑔 Fn 𝑧𝑧𝐴) → dom 𝑔𝐴)
76adantrr 716 . . . 4 ((𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)) → dom 𝑔𝐴)
873adant3 1129 . . 3 ((𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → dom 𝑔𝐴)
98exlimiv 1932 . 2 (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → dom 𝑔𝐴)
103, 9sylbi 220 1 (𝑔𝐵 → dom 𝑔𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wral 3133  wss 3919  dom cdm 5542  cres 5544  Predcpred 6134   Fn wfn 6338  cfv 6343  (class class class)co 7149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-ov 7152
This theorem is referenced by:  frrlem7  33186  fprlem1  33194  frrlem15  33199
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