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Theorem bnj1127 30313
 Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1127 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)

Proof of Theorem bnj1127
Dummy variables 𝑓 𝑖 𝑗 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 250 . 2 ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 biid 250 . 2 (∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 eqid 2610 . 2 (ω ∖ {∅}) = (ω ∖ {∅})
4 eqid 2610 . 2 {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} = {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))}
5 biid 250 . 2 ((𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
6 biid 250 . 2 (((𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓𝑖) ⊆ 𝐴) ↔ ((𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓𝑖) ⊆ 𝐴))
7 biid 250 . 2 (∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]((𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓𝑖) ⊆ 𝐴)) ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]((𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓𝑖) ⊆ 𝐴)))
8 biid 250 . 2 ([𝑗 / 𝑖](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ [𝑗 / 𝑖](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
9 biid 250 . 2 ([𝑗 / 𝑖]𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝑗 / 𝑖]𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
10 biid 250 . 2 ([𝑗 / 𝑖](𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [𝑗 / 𝑖](𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
11 biid 250 . 2 ([𝑗 / 𝑖]((𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓𝑖) ⊆ 𝐴) ↔ [𝑗 / 𝑖]((𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓𝑖) ⊆ 𝐴))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1128 30312 1 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  [wsbc 3402   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ ciun 4455   class class class wbr 4583   E cep 4947  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957   ∧ w-bnj17 30005   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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fn 5807  df-fv 5812  df-om 6958  df-bnj17 30006  df-bnj14 30008  df-bnj18 30014 This theorem is referenced by:  bnj1125  30314  bnj1136  30319  bnj1413  30357
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