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Theorem bnj1522 30239
 Description: Well-founded recursion, part 3 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
bnj1522.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1522.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1522.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1522.4 𝐹 = 𝐶
Assertion
Ref Expression
bnj1522 ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑥,𝐻   𝑅,𝑑,𝑓,𝑥   𝑌,𝑑
Allowed substitution hints:   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝐻(𝑓,𝑑)   𝑌(𝑥,𝑓)

Proof of Theorem bnj1522
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1522.1 . 2 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2 bnj1522.2 . 2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3 bnj1522.3 . 2 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
4 bnj1522.4 . 2 𝐹 = 𝐶
5 biid 249 . 2 ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ↔ (𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
6 biid 249 . 2 (((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ∧ 𝐹𝐻) ↔ ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ∧ 𝐹𝐻))
7 biid 249 . 2 ((((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ∧ 𝐹𝐻) ∧ 𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)) ↔ (((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ∧ 𝐹𝐻) ∧ 𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)))
8 eqid 2514 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}
9 biid 249 . 2 (((((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ∧ 𝐹𝐻) ∧ 𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)) ∧ 𝑦 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} ∧ ∀𝑧 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} ¬ 𝑧𝑅𝑦) ↔ ((((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ∧ 𝐹𝐻) ∧ 𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)) ∧ 𝑦 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} ∧ ∀𝑧 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} ¬ 𝑧𝑅𝑦))
101, 2, 3, 4, 5, 6, 7, 8, 9bnj1523 30238 1 ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   ∧ w3a 1030   = wceq 1474   ∈ wcel 1938  {cab 2500   ≠ wne 2684  ∀wral 2800  ∃wrex 2801  {crab 2804   ⊆ wss 3444  ⟨cop 4034  ∪ cuni 4270   class class class wbr 4481   ↾ cres 4934   Fn wfn 5684  ‘cfv 5689   predc-bnj14 29852   FrSe w-bnj15 29856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-reg 8256  ax-inf2 8297 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-om 6834  df-1o 7323  df-bnj17 29851  df-bnj14 29853  df-bnj13 29855  df-bnj15 29857  df-bnj18 29859  df-bnj19 29861 This theorem is referenced by: (None)
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