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Theorem bnj1500 31110
Description: Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1500.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1500.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1500.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1500.4 𝐹 = 𝐶
Assertion
Ref Expression
bnj1500 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑅,𝑑,𝑓,𝑥   𝑌,𝑑
Allowed substitution hints:   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓)

Proof of Theorem bnj1500
StepHypRef Expression
1 bnj1500.1 . 2 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2 bnj1500.2 . 2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3 bnj1500.3 . 2 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
4 bnj1500.4 . 2 𝐹 = 𝐶
5 biid 251 . 2 ((𝑅 FrSe 𝐴𝑥𝐴) ↔ (𝑅 FrSe 𝐴𝑥𝐴))
6 biid 251 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝑓𝐶𝑥 ∈ dom 𝑓) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝑓𝐶𝑥 ∈ dom 𝑓))
7 biid 251 . 2 ((((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝑓𝐶𝑥 ∈ dom 𝑓) ∧ 𝑑𝐵 ∧ dom 𝑓 = 𝑑) ↔ (((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝑓𝐶𝑥 ∈ dom 𝑓) ∧ 𝑑𝐵 ∧ dom 𝑓 = 𝑑))
81, 2, 3, 4, 5, 6, 7bnj1501 31109 1 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  {cab 2606  wral 2909  wrex 2910  wss 3567  cop 4174   cuni 4427  dom cdm 5104  cres 5106   Fn wfn 5871  cfv 5876   predc-bnj14 30728   FrSe w-bnj15 30732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-reg 8482  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-1o 7545  df-bnj17 30727  df-bnj14 30729  df-bnj13 30731  df-bnj15 30733  df-bnj18 30735  df-bnj19 30737
This theorem is referenced by:  bnj1523  31113
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