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Theorem dicvalrelN 40056
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 5822 . . . 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 2733 . . . . . . . . . 10 (leβ€˜πΎ) = (leβ€˜πΎ)
3 eqid 2733 . . . . . . . . . 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 40055 . . . . . . . . 9 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ dom 𝐼 = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ Β¬ 𝑝(leβ€˜πΎ)π‘Š})
76eleq2d 2820 . . . . . . . 8 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ Β¬ 𝑝(leβ€˜πΎ)π‘Š}))
8 breq1 5152 . . . . . . . . . 10 (𝑝 = 𝑋 β†’ (𝑝(leβ€˜πΎ)π‘Š ↔ 𝑋(leβ€˜πΎ)π‘Š))
98notbid 318 . . . . . . . . 9 (𝑝 = 𝑋 β†’ (Β¬ 𝑝(leβ€˜πΎ)π‘Š ↔ Β¬ 𝑋(leβ€˜πΎ)π‘Š))
109elrab 3684 . . . . . . . 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 287 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑋(leβ€˜πΎ)π‘Š)))
1211biimpa 478 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ (𝑋 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑋(leβ€˜πΎ)π‘Š))
13 eqid 2733 . . . . . . 7 ((ocβ€˜πΎ)β€˜π‘Š) = ((ocβ€˜πΎ)β€˜π‘Š)
14 eqid 2733 . . . . . . 7 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
15 eqid 2733 . . . . . . 7 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
162, 3, 4, 13, 14, 15, 5dicval 40047 . . . . . 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 592 . . . . 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 5779 . . . 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 258 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ Rel (πΌβ€˜π‘‹))
2019ex 414 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 β†’ Rel (πΌβ€˜π‘‹)))
21 rel0 5800 . . 3 Rel βˆ…
22 ndmfv 6927 . . . 4 (Β¬ 𝑋 ∈ dom 𝐼 β†’ (πΌβ€˜π‘‹) = βˆ…)
2322releqd 5779 . . 3 (Β¬ 𝑋 ∈ dom 𝐼 β†’ (Rel (πΌβ€˜π‘‹) ↔ Rel βˆ…))
2421, 23mpbiri 258 . 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 397   = wceq 1542   ∈ wcel 2107  {crab 3433  βˆ…c0 4323   class class class wbr 5149  {copab 5211  dom cdm 5677  Rel wrel 5682  β€˜cfv 6544  β„©crio 7364  lecple 17204  occoc 17205  Atomscatm 38133  LHypclh 38855  LTrncltrn 38972  TEndoctendo 39623  DIsoCcdic 40043
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-dic 40044
This theorem is referenced by: (None)
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