Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frrlem5e Structured version   Visualization version   GIF version

Theorem frrlem5e 31489
Description: Lemma for founded recursion. The domain of the union of a subset of 𝐵 is closed under predecessors. (Contributed by Paul Chapman, 1-May-2012.)
Hypotheses
Ref Expression
frrlem5.1 𝑅 Fr 𝐴
frrlem5.2 𝑅 Se 𝐴
frrlem5.3 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
Assertion
Ref Expression
frrlem5e (𝐶𝐵 → (𝑋 ∈ dom 𝐶 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝐵(𝑦,𝑓)   𝐶(𝑥,𝑦,𝑓)   𝑋(𝑥,𝑦,𝑓)

Proof of Theorem frrlem5e
Dummy variables 𝑧 𝑡 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmuni 5294 . . . 4 dom 𝐶 = 𝑧𝐶 dom 𝑧
21eleq2i 2690 . . 3 (𝑋 ∈ dom 𝐶𝑋 𝑧𝐶 dom 𝑧)
3 eliun 4490 . . 3 (𝑋 𝑧𝐶 dom 𝑧 ↔ ∃𝑧𝐶 𝑋 ∈ dom 𝑧)
42, 3bitri 264 . 2 (𝑋 ∈ dom 𝐶 ↔ ∃𝑧𝐶 𝑋 ∈ dom 𝑧)
5 ssel2 3578 . . . . 5 ((𝐶𝐵𝑧𝐶) → 𝑧𝐵)
6 frrlem5.3 . . . . . . . 8 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
76frrlem1 31481 . . . . . . 7 𝐵 = {𝑧 ∣ ∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))))}
87abeq2i 2732 . . . . . 6 (𝑧𝐵 ↔ ∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡))))))
9 fndm 5948 . . . . . . . . 9 (𝑧 Fn 𝑤 → dom 𝑧 = 𝑤)
10 predeq3 5643 . . . . . . . . . . . . 13 (𝑡 = 𝑋 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑋))
1110sseq1d 3611 . . . . . . . . . . . 12 (𝑡 = 𝑋 → (Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
1211rspccv 3292 . . . . . . . . . . 11 (∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 → (𝑋𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
13123ad2ant2 1081 . . . . . . . . . 10 ((𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
14 eleq2 2687 . . . . . . . . . . 11 (dom 𝑧 = 𝑤 → (𝑋 ∈ dom 𝑧𝑋𝑤))
15 sseq2 3606 . . . . . . . . . . 11 (dom 𝑧 = 𝑤 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
1614, 15imbi12d 334 . . . . . . . . . 10 (dom 𝑧 = 𝑤 → ((𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) ↔ (𝑋𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤)))
1713, 16syl5ibr 236 . . . . . . . . 9 (dom 𝑧 = 𝑤 → ((𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧)))
189, 17syl 17 . . . . . . . 8 (𝑧 Fn 𝑤 → ((𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧)))
1918imp 445 . . . . . . 7 ((𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡))))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
2019exlimiv 1855 . . . . . 6 (∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡))))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
218, 20sylbi 207 . . . . 5 (𝑧𝐵 → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
225, 21syl 17 . . . 4 ((𝐶𝐵𝑧𝐶) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
23 dmeq 5284 . . . . . . . . . 10 (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
2423sseq2d 3612 . . . . . . . . 9 (𝑤 = 𝑧 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
2524rspcev 3295 . . . . . . . 8 ((𝑧𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → ∃𝑤𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤)
26 ssiun 4528 . . . . . . . 8 (∃𝑤𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤𝐶 dom 𝑤)
2725, 26syl 17 . . . . . . 7 ((𝑧𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤𝐶 dom 𝑤)
28 dmuni 5294 . . . . . . 7 dom 𝐶 = 𝑤𝐶 dom 𝑤
2927, 28syl6sseqr 3631 . . . . . 6 ((𝑧𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶)
3029ex 450 . . . . 5 (𝑧𝐶 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
3130adantl 482 . . . 4 ((𝐶𝐵𝑧𝐶) → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
3222, 31syld 47 . . 3 ((𝐶𝐵𝑧𝐶) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
3332rexlimdva 3024 . 2 (𝐶𝐵 → (∃𝑧𝐶 𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
344, 33syl5bi 232 1 (𝐶𝐵 → (𝑋 ∈ dom 𝐶 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  wss 3555   cuni 4402   ciun 4485   Fr wfr 5030   Se wse 5031  dom cdm 5074  cres 5076  Predcpred 5638   Fn wfn 5842  cfv 5847  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-ov 6607
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator