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Theorem bnj1137 31189
Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1137.1 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
Assertion
Ref Expression
bnj1137 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo(𝐵, 𝐴, 𝑅))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑋
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem bnj1137
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 bnj1137.1 . . . . . 6 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
21eleq2i 2722 . . . . 5 (𝑣𝐵𝑣 ∈ ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
3 elun 3786 . . . . 5 (𝑣 ∈ ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ↔ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
42, 3bitri 264 . . . 4 (𝑣𝐵 ↔ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
5 bnj213 31078 . . . . . . . . 9 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
65sseli 3632 . . . . . . . 8 (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) → 𝑣𝐴)
7 bnj906 31126 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑣𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
87adantlr 751 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
96, 8sylan2 490 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
10 bnj906 31126 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
1110sselda 3636 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → 𝑣 ∈ trCl(𝑋, 𝐴, 𝑅))
12 bnj18eq1 31123 . . . . . . . . 9 (𝑦 = 𝑣 → trCl(𝑦, 𝐴, 𝑅) = trCl(𝑣, 𝐴, 𝑅))
1312ssiun2s 4596 . . . . . . . 8 (𝑣 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑣, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
1411, 13syl 17 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → trCl(𝑣, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
159, 14sstrd 3646 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
16 bnj1147 31188 . . . . . . . . . . 11 trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
1716rgenw 2953 . . . . . . . . . 10 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
18 iunss 4593 . . . . . . . . . 10 ( 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴 ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
1917, 18mpbir 221 . . . . . . . . 9 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
2019sseli 3632 . . . . . . . 8 (𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) → 𝑣𝐴)
2120, 8sylan2 490 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
22 bnj1125 31186 . . . . . . . . . . . 12 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
23223expia 1286 . . . . . . . . . . 11 ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
2423ralrimiv 2994 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25 iunss 4593 . . . . . . . . . 10 ( 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))
2624, 25sylibr 224 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
2726sselda 3636 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) → 𝑣 ∈ trCl(𝑋, 𝐴, 𝑅))
2827, 13syl 17 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑣, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
2921, 28sstrd 3646 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
3015, 29jaodan 843 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
31 ssun2 3810 . . . . . 6 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
3231, 1sseqtr4i 3671 . . . . 5 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐵
3330, 32syl6ss 3648 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
344, 33sylan2b 491 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣𝐵) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
3534ralrimiva 2995 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑣𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
36 df-bnj19 30891 . 2 ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑣𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
3735, 36sylibr 224 1 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo(𝐵, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  cun 3605  wss 3607   ciun 4552   predc-bnj14 30882   FrSe w-bnj15 30886   trClc-bnj18 30888   TrFow-bnj19 30890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-bnj17 30881  df-bnj14 30883  df-bnj13 30885  df-bnj15 30887  df-bnj18 30889  df-bnj19 30891
This theorem is referenced by:  bnj1136  31191
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