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Theorem bnj1015 30285
Description: Technical lemma for bnj69 30332. 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 3186 . 2 𝐽 ∈ V
3 eleq1 2676 . . . 4 (𝑗 = 𝐽 → (𝑗 ∈ dom 𝐺𝐽 ∈ dom 𝐺))
43anbi2d 736 . . 3 (𝑗 = 𝐽 → ((𝐺𝐵𝑗 ∈ dom 𝐺) ↔ (𝐺𝐵𝐽 ∈ dom 𝐺)))
5 fveq2 6103 . . . 4 (𝑗 = 𝐽 → (𝐺𝑗) = (𝐺𝐽))
65sseq1d 3595 . . 3 (𝑗 = 𝐽 → ((𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)))
74, 6imbi12d 333 . 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 3186 . . 3 𝐺 ∈ V
10 eleq1 2676 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐵𝐺𝐵))
11 dmeq 5246 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
1211eleq2d 2673 . . . . 5 (𝑔 = 𝐺 → (𝑗 ∈ dom 𝑔𝑗 ∈ dom 𝐺))
1310, 12anbi12d 743 . . . 4 (𝑔 = 𝐺 → ((𝑔𝐵𝑗 ∈ dom 𝑔) ↔ (𝐺𝐵𝑗 ∈ dom 𝐺)))
14 fveq1 6102 . . . . 5 (𝑔 = 𝐺 → (𝑔𝑗) = (𝐺𝑗))
1514sseq1d 3595 . . . 4 (𝑔 = 𝐺 → ((𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
1613, 15imbi12d 333 . . 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 30284 . . 3 ((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
229, 16, 21vtocl 3232 . 2 ((𝐺𝐵𝑗 ∈ dom 𝐺) → (𝐺𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
232, 7, 22vtocl 3232 1 ((𝐺𝐵𝐽 ∈ dom 𝐺) → (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  {cab 2596  wral 2896  wrex 2897  cdif 3537  wss 3540  c0 3874  {csn 4125   ciun 4455  dom cdm 5038  suc csuc 5642   Fn wfn 5799  cfv 5804  ωcom 6957   predc-bnj14 30007   trClc-bnj18 30013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-bnj18 30014
This theorem is referenced by:  bnj1018  30286
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