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Theorem tcel 8872
Description: The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
Hypothesis
Ref Expression
tc2.1 𝐴 ∈ V
Assertion
Ref Expression
tcel (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Proof of Theorem tcel
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tcvalg 8865 . 2 (𝐵𝐴 → (TC‘𝐵) = {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
2 ssel 3793 . . . . . . . 8 (𝐴𝑥 → (𝐵𝐴𝐵𝑥))
3 trss 4955 . . . . . . . . 9 (Tr 𝑥 → (𝐵𝑥𝐵𝑥))
43com12 32 . . . . . . . 8 (𝐵𝑥 → (Tr 𝑥𝐵𝑥))
52, 4syl6com 37 . . . . . . 7 (𝐵𝐴 → (𝐴𝑥 → (Tr 𝑥𝐵𝑥)))
65impd 399 . . . . . 6 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → 𝐵𝑥))
7 simpr 478 . . . . . . 7 ((𝐴𝑥 ∧ Tr 𝑥) → Tr 𝑥)
87a1i 11 . . . . . 6 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → Tr 𝑥))
96, 8jcad 509 . . . . 5 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → (𝐵𝑥 ∧ Tr 𝑥)))
109ss2abdv 3872 . . . 4 (𝐵𝐴 → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
11 intss 4689 . . . 4 ({𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} → {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1210, 11syl 17 . . 3 (𝐵𝐴 {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
13 tc2.1 . . . 4 𝐴 ∈ V
14 tcvalg 8865 . . . 4 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1513, 14ax-mp 5 . . 3 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
1612, 15syl6sseqr 3849 . 2 (𝐵𝐴 {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
171, 16eqsstrd 3836 1 (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {cab 2786  Vcvv 3386  wss 3770   cint 4668  Tr wtr 4946  cfv 6102  TCctc 8863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184  ax-inf2 8789
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-om 7301  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-tc 8864
This theorem is referenced by:  tcrank  8998  hsmexlem4  9540
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