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Theorem bnj1413 32415
 Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1413.1 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
Assertion
Ref Expression
bnj1413 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑋
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem bnj1413
StepHypRef Expression
1 bnj1148 32376 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)
2 bnj893 32308 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
3 simp1 1133 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
4 bnj1127 32371 . . . . . . . 8 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑦𝐴)
543ad2ant3 1132 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑦𝐴)
6 bnj893 32308 . . . . . . 7 ((𝑅 FrSe 𝐴𝑦𝐴) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
73, 5, 6syl2anc 587 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
873expia 1118 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ∈ V))
98ralrimiv 3151 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
10 iunexg 7650 . . . 4 (( 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)
112, 9, 10syl2anc 587 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
121, 11bnj1149 32172 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ∈ V)
13 bnj1413.1 . . 3 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
14 bnj906 32310 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
15 iunss1 4898 . . . 4 ( pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) → 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
16 unss2 4111 . . . 4 ( 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) → ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
1714, 15, 163syl 18 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
1813, 17eqsstrid 3966 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
1912, 18ssexd 5195 1 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ∀wral 3109  Vcvv 3444   ∪ cun 3882   ⊆ wss 3884  ∪ ciun 4884   predc-bnj14 32066   FrSe w-bnj15 32070   trClc-bnj18 32072 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-reg 9044  ax-inf2 9092 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1o 8089  df-bnj17 32065  df-bnj14 32067  df-bnj13 32069  df-bnj15 32071  df-bnj18 32073 This theorem is referenced by:  bnj1408  32416
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