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Theorem bnj1015 34626
Description: Technical lemma for bnj69 34674. 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 3493 . 2 𝐽 ∈ V
3 eleq1 2817 . . . 4 (𝑗 = 𝐽 → (𝑗 ∈ dom 𝐺𝐽 ∈ dom 𝐺))
43anbi2d 628 . . 3 (𝑗 = 𝐽 → ((𝐺𝐵𝑗 ∈ dom 𝐺) ↔ (𝐺𝐵𝐽 ∈ dom 𝐺)))
5 fveq2 6902 . . . 4 (𝑗 = 𝐽 → (𝐺𝑗) = (𝐺𝐽))
65sseq1d 4013 . . 3 (𝑗 = 𝐽 → ((𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)))
74, 6imbi12d 343 . 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 3493 . . 3 𝐺 ∈ V
10 eleq1 2817 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐵𝐺𝐵))
11 dmeq 5910 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
1211eleq2d 2815 . . . . 5 (𝑔 = 𝐺 → (𝑗 ∈ dom 𝑔𝑗 ∈ dom 𝐺))
1310, 12anbi12d 630 . . . 4 (𝑔 = 𝐺 → ((𝑔𝐵𝑗 ∈ dom 𝑔) ↔ (𝐺𝐵𝑗 ∈ dom 𝐺)))
14 fveq1 6901 . . . . 5 (𝑔 = 𝐺 → (𝑔𝑗) = (𝐺𝑗))
1514sseq1d 4013 . . . 4 (𝑔 = 𝐺 → ((𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
1613, 15imbi12d 343 . . 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 34625 . . 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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2705  wral 3058  wrex 3067  cdif 3946  wss 3949  c0 4326  {csn 4632   ciun 5000  dom cdm 5682  suc csuc 6376   Fn wfn 6548  cfv 6553  ωcom 7876   predc-bnj14 34352   trClc-bnj18 34358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-dm 5692  df-iota 6505  df-fv 6561  df-bnj18 34359
This theorem is referenced by:  bnj1018g  34627  bnj1018  34628
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