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Theorem dicvalrelN 40359
Description: The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicvalrel.h 𝐻 = (LHypβ€˜πΎ)
dicvalrel.i 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dicvalrelN ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Rel (πΌβ€˜π‘‹))

Proof of Theorem dicvalrelN
Dummy variables 𝑓 𝑔 𝑝 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopabv 5820 . . . 4 Rel {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑋)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))}
2 eqid 2730 . . . . . . . . . 10 (leβ€˜πΎ) = (leβ€˜πΎ)
3 eqid 2730 . . . . . . . . . 10 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 dicvalrel.h . . . . . . . . . 10 𝐻 = (LHypβ€˜πΎ)
5 dicvalrel.i . . . . . . . . . 10 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
62, 3, 4, 5dicdmN 40358 . . . . . . . . 9 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ dom 𝐼 = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ Β¬ 𝑝(leβ€˜πΎ)π‘Š})
76eleq2d 2817 . . . . . . . 8 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ Β¬ 𝑝(leβ€˜πΎ)π‘Š}))
8 breq1 5150 . . . . . . . . . 10 (𝑝 = 𝑋 β†’ (𝑝(leβ€˜πΎ)π‘Š ↔ 𝑋(leβ€˜πΎ)π‘Š))
98notbid 317 . . . . . . . . 9 (𝑝 = 𝑋 β†’ (Β¬ 𝑝(leβ€˜πΎ)π‘Š ↔ Β¬ 𝑋(leβ€˜πΎ)π‘Š))
109elrab 3682 . . . . . . . 8 (𝑋 ∈ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ Β¬ 𝑝(leβ€˜πΎ)π‘Š} ↔ (𝑋 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑋(leβ€˜πΎ)π‘Š))
117, 10bitrdi 286 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑋(leβ€˜πΎ)π‘Š)))
1211biimpa 475 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ (𝑋 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑋(leβ€˜πΎ)π‘Š))
13 eqid 2730 . . . . . . 7 ((ocβ€˜πΎ)β€˜π‘Š) = ((ocβ€˜πΎ)β€˜π‘Š)
14 eqid 2730 . . . . . . 7 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
15 eqid 2730 . . . . . . 7 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
162, 3, 4, 13, 14, 15, 5dicval 40350 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑋(leβ€˜πΎ)π‘Š)) β†’ (πΌβ€˜π‘‹) = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑋)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))})
1712, 16syldan 589 . . . . 5 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ (πΌβ€˜π‘‹) = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑋)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))})
1817releqd 5777 . . . 4 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ (Rel (πΌβ€˜π‘‹) ↔ Rel {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑋)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))}))
191, 18mpbiri 257 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ Rel (πΌβ€˜π‘‹))
2019ex 411 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 β†’ Rel (πΌβ€˜π‘‹)))
21 rel0 5798 . . 3 Rel βˆ…
22 ndmfv 6925 . . . 4 (Β¬ 𝑋 ∈ dom 𝐼 β†’ (πΌβ€˜π‘‹) = βˆ…)
2322releqd 5777 . . 3 (Β¬ 𝑋 ∈ dom 𝐼 β†’ (Rel (πΌβ€˜π‘‹) ↔ Rel βˆ…))
2421, 23mpbiri 257 . 2 (Β¬ 𝑋 ∈ dom 𝐼 β†’ Rel (πΌβ€˜π‘‹))
2520, 24pm2.61d1 180 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Rel (πΌβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {crab 3430  βˆ…c0 4321   class class class wbr 5147  {copab 5209  dom cdm 5675  Rel wrel 5680  β€˜cfv 6542  β„©crio 7366  lecple 17208  occoc 17209  Atomscatm 38436  LHypclh 39158  LTrncltrn 39275  TEndoctendo 39926  DIsoCcdic 40346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-dic 40347
This theorem is referenced by: (None)
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