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Theorem bnj916 30711
Description: Technical lemma for bnj69 30786. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj916.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj916.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj916.3 𝐷 = (ω ∖ {∅})
bnj916.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj916.5 (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))
Assertion
Ref Expression
bnj916 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑛)   𝐷(𝑦,𝑓,𝑛)

Proof of Theorem bnj916
StepHypRef Expression
1 bnj256 30479 . . . . . 6 ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ (𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))))
212exbii 1772 . . . . 5 (∃𝑛𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ ∃𝑛𝑖((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ (𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))))
3 19.41v 1911 . . . . . 6 (∃𝑛((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))) ↔ (∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))))
4 nfv 1840 . . . . . . . . 9 𝑖 𝑛𝐷
5 bnj916.1 . . . . . . . . . . 11 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
6 bnj916.2 . . . . . . . . . . 11 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
75, 6bnj911 30710 . . . . . . . . . 10 ((𝑓 Fn 𝑛𝜑𝜓) → ∀𝑖(𝑓 Fn 𝑛𝜑𝜓))
87nf5i 2021 . . . . . . . . 9 𝑖(𝑓 Fn 𝑛𝜑𝜓)
94, 8nfan 1825 . . . . . . . 8 𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓))
10919.42 2103 . . . . . . 7 (∃𝑖((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ (𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))))
1110exbii 1771 . . . . . 6 (∃𝑛𝑖((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ (𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))) ↔ ∃𝑛((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))))
12 df-rex 2913 . . . . . . 7 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
13 df-rex 2913 . . . . . . 7 (∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖) ↔ ∃𝑖(𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)))
1412, 13anbi12i 732 . . . . . 6 ((∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)) ↔ (∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))))
153, 11, 143bitr4i 292 . . . . 5 (∃𝑛𝑖((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ (𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖))) ↔ (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)))
162, 15bitri 264 . . . 4 (∃𝑛𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)))
1716exbii 1771 . . 3 (∃𝑓𝑛𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ ∃𝑓(∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)))
18 bnj916.5 . . . . . . 7 (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))
19183anbi2i 1252 . . . . . 6 ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓))
2019anbi1i 730 . . . . 5 (((𝑛𝐷𝜒𝑖 ∈ dom 𝑓) ∧ 𝑦 ∈ (𝑓𝑖)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓) ∧ 𝑦 ∈ (𝑓𝑖)))
21 df-bnj17 30460 . . . . 5 ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓) ∧ 𝑦 ∈ (𝑓𝑖)))
22 df-bnj17 30460 . . . . 5 ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓) ∧ 𝑦 ∈ (𝑓𝑖)))
2320, 21, 223bitr4i 292 . . . 4 ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)))
24233exbii 1773 . . 3 (∃𝑓𝑛𝑖(𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ ∃𝑓𝑛𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)))
25 bnj916.3 . . . . . . 7 𝐷 = (ω ∖ {∅})
26 bnj916.4 . . . . . . 7 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
275, 6, 25, 26bnj882 30704 . . . . . 6 trCl(𝑋, 𝐴, 𝑅) = 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖)
2827eleq2i 2690 . . . . 5 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑦 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖))
29 eliun 4490 . . . . 5 (𝑦 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ↔ ∃𝑓𝐵 𝑦 𝑖 ∈ dom 𝑓(𝑓𝑖))
30 eliun 4490 . . . . . 6 (𝑦 𝑖 ∈ dom 𝑓(𝑓𝑖) ↔ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖))
3130rexbii 3034 . . . . 5 (∃𝑓𝐵 𝑦 𝑖 ∈ dom 𝑓(𝑓𝑖) ↔ ∃𝑓𝐵𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖))
3228, 29, 313bitri 286 . . . 4 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓𝐵𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖))
33 df-rex 2913 . . . 4 (∃𝑓𝐵𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖) ↔ ∃𝑓(𝑓𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)))
3426abeq2i 2732 . . . . . 6 (𝑓𝐵 ↔ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓))
3534anbi1i 730 . . . . 5 ((𝑓𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)) ↔ (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)))
3635exbii 1771 . . . 4 (∃𝑓(𝑓𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)) ↔ ∃𝑓(∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)))
3732, 33, 363bitri 286 . . 3 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓(∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓𝑖)))
3817, 24, 373bitr4ri 293 . 2 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓𝑛𝑖(𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)))
39 bnj643 30527 . . . . . 6 ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) → 𝜒)
4018bnj564 30522 . . . . . . 7 (𝜒 → dom 𝑓 = 𝑛)
4140eleq2d 2684 . . . . . 6 (𝜒 → (𝑖 ∈ dom 𝑓𝑖𝑛))
42 anbi1 742 . . . . . . 7 ((𝑖 ∈ dom 𝑓𝑖𝑛) → ((𝑖 ∈ dom 𝑓 ∧ (𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖))) ↔ (𝑖𝑛 ∧ (𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖)))))
43 bnj334 30486 . . . . . . . 8 ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ (𝑖 ∈ dom 𝑓𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖)))
44 bnj252 30476 . . . . . . . 8 ((𝑖 ∈ dom 𝑓𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖)) ↔ (𝑖 ∈ dom 𝑓 ∧ (𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖))))
4543, 44bitri 264 . . . . . . 7 ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ (𝑖 ∈ dom 𝑓 ∧ (𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖))))
46 bnj334 30486 . . . . . . . 8 ((𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ (𝑖𝑛𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖)))
47 bnj252 30476 . . . . . . . 8 ((𝑖𝑛𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖)) ↔ (𝑖𝑛 ∧ (𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖))))
4846, 47bitri 264 . . . . . . 7 ((𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ (𝑖𝑛 ∧ (𝑛𝐷𝜒𝑦 ∈ (𝑓𝑖))))
4942, 45, 483bitr4g 303 . . . . . 6 ((𝑖 ∈ dom 𝑓𝑖𝑛) → ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ (𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖))))
5039, 41, 493syl 18 . . . . 5 ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) → ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) ↔ (𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖))))
5150ibi 256 . . . 4 ((𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) → (𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
52512eximi 1760 . . 3 (∃𝑛𝑖(𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) → ∃𝑛𝑖(𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
5352eximi 1759 . 2 (∃𝑓𝑛𝑖(𝑛𝐷𝜒𝑖 ∈ dom 𝑓𝑦 ∈ (𝑓𝑖)) → ∃𝑓𝑛𝑖(𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
5438, 53sylbi 207 1 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  cdif 3552  c0 3891  {csn 4148   ciun 4485  dom cdm 5074  suc csuc 5684   Fn wfn 5842  cfv 5847  ωcom 7012  w-bnj17 30459   predc-bnj14 30461   trClc-bnj18 30467
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-v 3188  df-iun 4487  df-fn 5850  df-bnj17 30460  df-bnj18 30468
This theorem is referenced by:  bnj917  30712
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