Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1136 Structured version   Visualization version   GIF version

Theorem bnj1136 30319
 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
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 217 . . 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 30318 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)
5 bnj893 30252 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
6 simp1 1054 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
7 bnj1127 30313 . . . . . . . . . . 11 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑦𝐴)
873ad2ant3 1077 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑦𝐴)
9 bnj893 30252 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑦𝐴) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
106, 8, 9syl2anc 691 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
11103expia 1259 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ∈ V))
1211ralrimiv 2948 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
13 iunexg 7035 . . . . . . 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 691 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
154, 14bnj1149 30117 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ∈ V)
163, 15syl5eqel 2692 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)
173bnj1137 30317 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo(𝐵, 𝐴, 𝑅))
183bnj931 30095 . . . . 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 1239 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝜏)
221, 20bnj1124 30310 . . 3 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
232, 21, 22syl2anc 691 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
24 bnj906 30254 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25 bnj1125 30314 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
26253expia 1259 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
2726ralrimiv 2948 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
28 ss2iun 4472 . . . . . 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 30115 . . . . . 6 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)
3028, 29syl6ss 3580 . . . . 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 3751 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ⊆ trCl(𝑋, 𝐴, 𝑅))
333, 32syl5eqss 3612 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ⊆ trCl(𝑋, 𝐴, 𝑅))
3423, 33eqssd 3585 1 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∪ ciun 4455   predc-bnj14 30007   FrSe w-bnj15 30011   trClc-bnj18 30013   TrFow-bnj19 30015 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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421 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-reu 2903  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-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016 This theorem is referenced by:  bnj1408  30358
 Copyright terms: Public domain W3C validator