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Theorem bnj60 32352
Description: Well-founded recursion, part 1 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
bnj60.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj60.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj60.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj60.4 𝐹 = 𝐶
Assertion
Ref Expression
bnj60 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑅,𝑑,𝑓,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj60
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj60.1 . . . . 5 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2 bnj60.2 . . . . 5 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3 bnj60.3 . . . . 5 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
41, 2, 3bnj1497 32350 . . . 4 𝑔𝐶 Fun 𝑔
5 eqid 2824 . . . . . . . 8 (dom 𝑔 ∩ dom ) = (dom 𝑔 ∩ dom )
61, 2, 3, 5bnj1311 32314 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
763expia 1118 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶) → (𝐶 → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))))
87ralrimiv 3175 . . . . 5 ((𝑅 FrSe 𝐴𝑔𝐶) → ∀𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
98ralrimiva 3176 . . . 4 (𝑅 FrSe 𝐴 → ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
10 biid 264 . . . . 5 (∀𝑔𝐶 Fun 𝑔 ↔ ∀𝑔𝐶 Fun 𝑔)
11 biid 264 . . . . 5 ((∀𝑔𝐶 Fun 𝑔 ∧ ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))) ↔ (∀𝑔𝐶 Fun 𝑔 ∧ ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))))
1210, 5, 11bnj1383 32121 . . . 4 ((∀𝑔𝐶 Fun 𝑔 ∧ ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))) → Fun 𝐶)
134, 9, 12sylancr 590 . . 3 (𝑅 FrSe 𝐴 → Fun 𝐶)
14 bnj60.4 . . . 4 𝐹 = 𝐶
1514funeqi 6357 . . 3 (Fun 𝐹 ↔ Fun 𝐶)
1613, 15sylibr 237 . 2 (𝑅 FrSe 𝐴 → Fun 𝐹)
171, 2, 3, 14bnj1498 32351 . 2 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
1816, 17bnj1422 32127 1 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {cab 2802  wral 3132  wrex 3133  cin 3917  wss 3918  cop 4554   cuni 4819  dom cdm 5536  cres 5538  Fun wfun 6330   Fn wfn 6331  cfv 6336   predc-bnj14 31976   FrSe w-bnj15 31980
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  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-reg 9040  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7564  df-1o 8085  df-bnj17 31975  df-bnj14 31977  df-bnj13 31979  df-bnj15 31981  df-bnj18 31983  df-bnj19 31985
This theorem is referenced by:  bnj1501  32357  bnj1523  32361
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