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Theorem bnj983 31349
Description: Technical lemma for bnj69 31406. 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
bnj983.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj983.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj983.3 𝐷 = (ω ∖ {∅})
bnj983.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj983.5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
Assertion
Ref Expression
bnj983 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝑓,𝑍,𝑖,𝑛   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)   𝑍(𝑦)

Proof of Theorem bnj983
StepHypRef Expression
1 bnj983.1 . . . . . . . 8 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj983.2 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj983.3 . . . . . . . 8 𝐷 = (ω ∖ {∅})
4 bnj983.4 . . . . . . . 8 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
51, 2, 3, 4bnj882 31324 . . . . . . 7 trCl(𝑋, 𝐴, 𝑅) = 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖)
65eleq2i 2884 . . . . . 6 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑍 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖))
7 eliun 4723 . . . . . . 7 (𝑍 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ↔ ∃𝑓𝐵 𝑍 𝑖 ∈ dom 𝑓(𝑓𝑖))
8 eliun 4723 . . . . . . . 8 (𝑍 𝑖 ∈ dom 𝑓(𝑓𝑖) ↔ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖))
98rexbii 3236 . . . . . . 7 (∃𝑓𝐵 𝑍 𝑖 ∈ dom 𝑓(𝑓𝑖) ↔ ∃𝑓𝐵𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖))
107, 9bitri 266 . . . . . 6 (𝑍 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ↔ ∃𝑓𝐵𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖))
11 df-rex 3109 . . . . . . 7 (∃𝑓𝐵𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖) ↔ ∃𝑓(𝑓𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖)))
124abeq2i 2926 . . . . . . . . 9 (𝑓𝐵 ↔ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓))
1312anbi1i 612 . . . . . . . 8 ((𝑓𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖)) ↔ (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖)))
1413exbii 1933 . . . . . . 7 (∃𝑓(𝑓𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖)) ↔ ∃𝑓(∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖)))
1511, 14bitri 266 . . . . . 6 (∃𝑓𝐵𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖) ↔ ∃𝑓(∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖)))
166, 10, 153bitri 288 . . . . 5 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓(∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖)))
17 bnj983.5 . . . . . . . . 9 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
18 bnj252 31100 . . . . . . . . 9 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
1917, 18bitri 266 . . . . . . . 8 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
2019exbii 1933 . . . . . . 7 (∃𝑛𝜒 ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
2120anbi1i 612 . . . . . 6 ((∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))) ↔ (∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
22 df-rex 3109 . . . . . . 7 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
23 df-rex 3109 . . . . . . 7 (∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖) ↔ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)))
2422, 23anbi12i 614 . . . . . 6 ((∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖)) ↔ (∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
2521, 24bitr4i 269 . . . . 5 ((∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))) ↔ (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓𝑖)))
2616, 25bnj133 31124 . . . 4 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓(∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
27 19.41v 2040 . . . 4 (∃𝑛(𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))) ↔ (∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
2826, 27bnj133 31124 . . 3 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓𝑛(𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
292bnj1095 31180 . . . . . . 7 (𝜓 → ∀𝑖𝜓)
3029, 17bnj1096 31181 . . . . . 6 (𝜒 → ∀𝑖𝜒)
3130nf5i 2191 . . . . 5 𝑖𝜒
323119.42 2274 . . . 4 (∃𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))) ↔ (𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
33322exbii 1934 . . 3 (∃𝑓𝑛𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))) ↔ ∃𝑓𝑛(𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
3428, 33bitr4i 269 . 2 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓𝑛𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
35 3anass 1109 . . 3 ((𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) ↔ (𝜒 ∧ (𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
36353exbii 1935 . 2 (∃𝑓𝑛𝑖(𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) ↔ ∃𝑓𝑛𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖))))
37 fndm 6204 . . . . . . . 8 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
3817, 37bnj770 31161 . . . . . . 7 (𝜒 → dom 𝑓 = 𝑛)
39 eleq2 2881 . . . . . . . 8 (dom 𝑓 = 𝑛 → (𝑖 ∈ dom 𝑓𝑖𝑛))
40393anbi2d 1558 . . . . . . 7 (dom 𝑓 = 𝑛 → ((𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) ↔ (𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))))
4138, 40syl 17 . . . . . 6 (𝜒 → ((𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) ↔ (𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))))
42413ad2ant1 1156 . . . . 5 ((𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) → ((𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) ↔ (𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))))
4342ibi 258 . . . 4 ((𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) → (𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
44413ad2ant1 1156 . . . . 5 ((𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)) → ((𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) ↔ (𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))))
4544ibir 259 . . . 4 ((𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)) → (𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)))
4643, 45impbii 200 . . 3 ((𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) ↔ (𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
47463exbii 1935 . 2 (∃𝑓𝑛𝑖(𝜒𝑖 ∈ dom 𝑓𝑍 ∈ (𝑓𝑖)) ↔ ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
4834, 36, 473bitr2i 290 1 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2157  {cab 2799  wral 3103  wrex 3104  cdif 3773  c0 4123  {csn 4377   ciun 4719  dom cdm 5318  suc csuc 5945   Fn wfn 6099  cfv 6104  ωcom 7298  w-bnj17 31083   predc-bnj14 31085   trClc-bnj18 31091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-v 3400  df-iun 4721  df-fn 6107  df-bnj17 31084  df-bnj18 31092
This theorem is referenced by:  bnj1033  31365
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