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Theorem bnj1413 31431
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 31392 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)
2 bnj893 31326 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
3 simp1 1159 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
4 bnj1127 31387 . . . . . . . 8 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑦𝐴)
543ad2ant3 1158 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑦𝐴)
6 bnj893 31326 . . . . . . 7 ((𝑅 FrSe 𝐴𝑦𝐴) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
73, 5, 6syl2anc 575 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
873expia 1143 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ∈ V))
98ralrimiv 3160 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
10 iunexg 7376 . . . 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 575 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
121, 11bnj1149 31191 . 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 31328 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
15 iunss1 4731 . . . 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 3990 . . . 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, 17syl5eqss 3853 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
1912, 18ssexd 5007 1 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2157  wral 3103  Vcvv 3398  cun 3774  wss 3776   ciun 4719   predc-bnj14 31085   FrSe w-bnj15 31089   trClc-bnj18 31091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-reg 8739  ax-inf2 8788
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-om 7299  df-1o 7799  df-bnj17 31084  df-bnj14 31086  df-bnj13 31088  df-bnj15 31090  df-bnj18 31092
This theorem is referenced by:  bnj1408  31432
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