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Theorem diatrl 39557
Description: Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)
Hypotheses
Ref Expression
diatrl.b 𝐡 = (Baseβ€˜πΎ)
diatrl.l ≀ = (leβ€˜πΎ)
diatrl.h 𝐻 = (LHypβ€˜πΎ)
diatrl.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diatrl.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
diatrl.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diatrl (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ (π‘…β€˜πΉ) ≀ 𝑋)

Proof of Theorem diatrl
StepHypRef Expression
1 diatrl.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 diatrl.l . . . 4 ≀ = (leβ€˜πΎ)
3 diatrl.h . . . 4 𝐻 = (LHypβ€˜πΎ)
4 diatrl.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 diatrl.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
6 diatrl.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
71, 2, 3, 4, 5, 6diaelval 39546 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) ↔ (𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋)))
8 simpr 486 . . 3 ((𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋) β†’ (π‘…β€˜πΉ) ≀ 𝑋)
97, 8syl6bi 253 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) β†’ (π‘…β€˜πΉ) ≀ 𝑋))
1093impia 1118 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ (π‘…β€˜πΉ) ≀ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5109  β€˜cfv 6500  Basecbs 17091  lecple 17148  LHypclh 38497  LTrncltrn 38614  trLctrl 38671  DIsoAcdia 39541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-disoa 39542
This theorem is referenced by:  dialss  39559  dibelval1st2N  39664  diblss  39683
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