Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1015 Structured version   Visualization version   GIF version

Theorem bnj1015 33631
Description: Technical lemma for bnj69 33679. 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
bnj1015.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1015.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1015.13 𝐷 = (ω ∖ {∅})
bnj1015.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1015.15 𝐺𝑉
bnj1015.16 𝐽𝑉
Assertion
Ref Expression
bnj1015 ((𝐺𝐵𝐽 ∈ dom 𝐺) → (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))
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)   𝐵(𝑦,𝑓,𝑖,𝑛)   𝐷(𝑦,𝑓,𝑛)   𝐺(𝑦,𝑓,𝑖,𝑛)   𝐽(𝑦,𝑓,𝑖,𝑛)   𝑉(𝑦,𝑓,𝑖,𝑛)

Proof of Theorem bnj1015
Dummy variables 𝑔 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1015.16 . . 3 𝐽𝑉
21elexi 3463 . 2 𝐽 ∈ V
3 eleq1 2822 . . . 4 (𝑗 = 𝐽 → (𝑗 ∈ dom 𝐺𝐽 ∈ dom 𝐺))
43anbi2d 630 . . 3 (𝑗 = 𝐽 → ((𝐺𝐵𝑗 ∈ dom 𝐺) ↔ (𝐺𝐵𝐽 ∈ dom 𝐺)))
5 fveq2 6843 . . . 4 (𝑗 = 𝐽 → (𝐺𝑗) = (𝐺𝐽))
65sseq1d 3976 . . 3 (𝑗 = 𝐽 → ((𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)))
74, 6imbi12d 345 . 2 (𝑗 = 𝐽 → (((𝐺𝐵𝑗 ∈ dom 𝐺) → (𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝐺𝐵𝐽 ∈ dom 𝐺) → (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))))
8 bnj1015.15 . . . 4 𝐺𝑉
98elexi 3463 . . 3 𝐺 ∈ V
10 eleq1 2822 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐵𝐺𝐵))
11 dmeq 5860 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
1211eleq2d 2820 . . . . 5 (𝑔 = 𝐺 → (𝑗 ∈ dom 𝑔𝑗 ∈ dom 𝐺))
1310, 12anbi12d 632 . . . 4 (𝑔 = 𝐺 → ((𝑔𝐵𝑗 ∈ dom 𝑔) ↔ (𝐺𝐵𝑗 ∈ dom 𝐺)))
14 fveq1 6842 . . . . 5 (𝑔 = 𝐺 → (𝑔𝑗) = (𝐺𝑗))
1514sseq1d 3976 . . . 4 (𝑔 = 𝐺 → ((𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
1613, 15imbi12d 345 . . 3 (𝑔 = 𝐺 → (((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝐺𝐵𝑗 ∈ dom 𝐺) → (𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))))
17 bnj1015.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
18 bnj1015.2 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
19 bnj1015.13 . . . 4 𝐷 = (ω ∖ {∅})
20 bnj1015.14 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
2117, 18, 19, 20bnj1014 33630 . . 3 ((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
229, 16, 21vtocl 3517 . 2 ((𝐺𝐵𝑗 ∈ dom 𝐺) → (𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
232, 7, 22vtocl 3517 1 ((𝐺𝐵𝐽 ∈ dom 𝐺) → (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  cdif 3908  wss 3911  c0 4283  {csn 4587   ciun 4955  dom cdm 5634  suc csuc 6320   Fn wfn 6492  cfv 6497  ωcom 7803   predc-bnj14 33357   trClc-bnj18 33363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-bnj18 33364
This theorem is referenced by:  bnj1018g  33632  bnj1018  33633
  Copyright terms: Public domain W3C validator