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Theorem wfrlem1 7583
Description: Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions 𝐵. This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 21-Apr-2011.)
Hypothesis
Ref Expression
wfrlem1.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Assertion
Ref Expression
wfrlem1 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
Distinct variable groups:   𝐴,𝑓,𝑔,𝑤,𝑥,𝑦,𝑧   𝑓,𝐹,𝑔,𝑤,𝑥,𝑦,𝑧   𝑅,𝑓,𝑔,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔)

Proof of Theorem wfrlem1
StepHypRef Expression
1 wfrlem1.1 . 2 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2 fneq1 6140 . . . . . 6 (𝑓 = 𝑔 → (𝑓 Fn 𝑥𝑔 Fn 𝑥))
3 fveq1 6351 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
4 reseq1 5545 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))
54fveq2d 6356 . . . . . . . 8 (𝑓 = 𝑔 → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))
63, 5eqeq12d 2775 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
76ralbidv 3124 . . . . . 6 (𝑓 = 𝑔 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
82, 73anbi13d 1550 . . . . 5 (𝑓 = 𝑔 → ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
98exbidv 1999 . . . 4 (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
10 fneq2 6141 . . . . . 6 (𝑥 = 𝑧 → (𝑔 Fn 𝑥𝑔 Fn 𝑧))
11 sseq1 3767 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
12 sseq2 3768 . . . . . . . . 9 (𝑥 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
1312raleqbi1dv 3285 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
14 predeq3 5845 . . . . . . . . . 10 (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
1514sseq1d 3773 . . . . . . . . 9 (𝑦 = 𝑤 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
1615cbvralv 3310 . . . . . . . 8 (∀𝑦𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
1713, 16syl6bb 276 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
1811, 17anbi12d 749 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))
19 raleq 3277 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑧 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
20 fveq2 6352 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑔𝑦) = (𝑔𝑤))
2114reseq2d 5551 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))
2221fveq2d 6356 . . . . . . . . 9 (𝑦 = 𝑤 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
2320, 22eqeq12d 2775 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
2423cbvralv 3310 . . . . . . 7 (∀𝑦𝑧 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
2519, 24syl6bb 276 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
2610, 18, 253anbi123d 1548 . . . . 5 (𝑥 = 𝑧 → ((𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
2726cbvexv 2420 . . . 4 (∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
289, 27syl6bb 276 . . 3 (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
2928cbvabv 2885 . 2 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
301, 29eqtri 2782 1 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1072   = wceq 1632  wex 1853  {cab 2746  wral 3050  wss 3715  cres 5268  Predcpred 5840   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  wfrlem2  7584  wfrlem3  7585  wfrlem3a  7586  wfrlem4  7587  wfrdmcl  7592
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