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Theorem bnj1136 34661
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
bnj1136.1 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
bnj1136.2 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
bnj1136.3 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
Assertion
Ref Expression
bnj1136 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑋
Allowed substitution hints:   𝜃(𝑦)   𝜏(𝑦)   𝐵(𝑦)

Proof of Theorem bnj1136
StepHypRef Expression
1 bnj1136.2 . . . 4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
21biimpri 227 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝜃)
3 bnj1136.1 . . . . 5 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
4 bnj1148 34660 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)
5 bnj893 34592 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
6 simp1 1133 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
7 bnj1127 34655 . . . . . . . . . . 11 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑦𝐴)
873ad2ant3 1132 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑦𝐴)
9 bnj893 34592 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑦𝐴) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
106, 8, 9syl2anc 582 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
11103expia 1118 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ∈ V))
1211ralrimiv 3142 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
13 iunexg 7973 . . . . . . 7 (( trCl(𝑋, 𝐴, 𝑅) ∈ V ∧ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
145, 12, 13syl2anc 582 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
154, 14bnj1149 34456 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ∈ V)
163, 15eqeltrid 2833 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)
173bnj1137 34659 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo(𝐵, 𝐴, 𝑅))
183bnj931 34434 . . . . 5 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵
1918a1i 11 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
20 bnj1136.3 . . . 4 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
2116, 17, 19, 20syl3anbrc 1340 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝜏)
221, 20bnj1124 34652 . . 3 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
232, 21, 22syl2anc 582 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
24 bnj906 34594 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25 bnj1125 34656 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
26253expia 1118 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
2726ralrimiv 3142 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
28 ss2iun 5018 . . . . . 6 (∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑋, 𝐴, 𝑅))
29 bnj1143 34454 . . . . . 6 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)
3028, 29sstrdi 3994 . . . . 5 (∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
3127, 30syl 17 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
3224, 31unssd 4188 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ⊆ trCl(𝑋, 𝐴, 𝑅))
333, 32eqsstrid 4030 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ⊆ trCl(𝑋, 𝐴, 𝑅))
3423, 33eqssd 3999 1 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  Vcvv 3473  cun 3947  wss 3949   ciun 5000   predc-bnj14 34352   FrSe w-bnj15 34356   trClc-bnj18 34358   TrFow-bnj19 34360
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  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9623  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-1o 8493  df-bnj17 34351  df-bnj14 34353  df-bnj13 34355  df-bnj15 34357  df-bnj18 34359  df-bnj19 34361
This theorem is referenced by:  bnj1408  34700
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