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Theorem bnj1015 32261
Description: Technical lemma for bnj69 32309. 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 3499 . 2 𝐽 ∈ V
3 eleq1 2903 . . . 4 (𝑗 = 𝐽 → (𝑗 ∈ dom 𝐺𝐽 ∈ dom 𝐺))
43anbi2d 631 . . 3 (𝑗 = 𝐽 → ((𝐺𝐵𝑗 ∈ dom 𝐺) ↔ (𝐺𝐵𝐽 ∈ dom 𝐺)))
5 fveq2 6659 . . . 4 (𝑗 = 𝐽 → (𝐺𝑗) = (𝐺𝐽))
65sseq1d 3984 . . 3 (𝑗 = 𝐽 → ((𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)))
74, 6imbi12d 348 . 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 3499 . . 3 𝐺 ∈ V
10 eleq1 2903 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐵𝐺𝐵))
11 dmeq 5760 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
1211eleq2d 2901 . . . . 5 (𝑔 = 𝐺 → (𝑗 ∈ dom 𝑔𝑗 ∈ dom 𝐺))
1310, 12anbi12d 633 . . . 4 (𝑔 = 𝐺 → ((𝑔𝐵𝑗 ∈ dom 𝑔) ↔ (𝐺𝐵𝑗 ∈ dom 𝐺)))
14 fveq1 6658 . . . . 5 (𝑔 = 𝐺 → (𝑔𝑗) = (𝐺𝑗))
1514sseq1d 3984 . . . 4 (𝑔 = 𝐺 → ((𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
1613, 15imbi12d 348 . . 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 32260 . . 3 ((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
229, 16, 21vtocl 3545 . 2 ((𝐺𝐵𝑗 ∈ dom 𝐺) → (𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
232, 7, 22vtocl 3545 1 ((𝐺𝐵𝐽 ∈ dom 𝐺) → (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  {cab 2802  wral 3133  wrex 3134  cdif 3916  wss 3919  c0 4276  {csn 4550   ciun 4906  dom cdm 5543  suc csuc 6181   Fn wfn 6339  cfv 6344  ωcom 7571   predc-bnj14 31985   trClc-bnj18 31991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-dm 5553  df-iota 6303  df-fv 6352  df-bnj18 31992
This theorem is referenced by:  bnj1018g  32262  bnj1018  32263
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