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Theorem bnj1137 33607
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 2829 . . . . 5 (𝑣𝐵𝑣 ∈ ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
3 elun 4108 . . . . 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 274 . . . 4 (𝑣𝐵 ↔ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
5 bnj213 33494 . . . . . . . . 9 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
65sseli 3940 . . . . . . . 8 (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) → 𝑣𝐴)
7 bnj906 33542 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑣𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
87adantlr 713 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
96, 8sylan2 593 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
10 bnj906 33542 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
1110sselda 3944 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → 𝑣 ∈ trCl(𝑋, 𝐴, 𝑅))
12 bnj18eq1 33539 . . . . . . . . 9 (𝑦 = 𝑣 → trCl(𝑦, 𝐴, 𝑅) = trCl(𝑣, 𝐴, 𝑅))
1312ssiun2s 5008 . . . . . . . 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 3954 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
16 bnj1147 33606 . . . . . . . . . . 11 trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
1716rgenw 3068 . . . . . . . . . 10 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
18 iunss 5005 . . . . . . . . . 10 ( 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴 ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
1917, 18mpbir 230 . . . . . . . . 9 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
2019sseli 3940 . . . . . . . 8 (𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) → 𝑣𝐴)
2120, 8sylan2 593 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
22 bnj1125 33604 . . . . . . . . . . . 12 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
23223expia 1121 . . . . . . . . . . 11 ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
2423ralrimiv 3142 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25 iunss 5005 . . . . . . . . . 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 233 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
2726sselda 3944 . . . . . . . 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 3954 . . . . . 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 956 . . . . 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 4133 . . . . . 6 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
3231, 1sseqtrri 3981 . . . . 5 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐵
3330, 32sstrdi 3956 . . . 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 594 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑣𝐵) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
3534ralrimiva 3143 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑣𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
36 df-bnj19 33309 . 2 ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑣𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
3735, 36sylibr 233 1 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo(𝐵, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3064  cun 3908  wss 3910   ciun 4954   predc-bnj14 33300   FrSe w-bnj15 33304   trClc-bnj18 33306   TrFow-bnj19 33308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-bnj17 33299  df-bnj14 33301  df-bnj13 33303  df-bnj15 33305  df-bnj18 33307  df-bnj19 33309
This theorem is referenced by:  bnj1136  33609
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