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Theorem diafn 40399
Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)
Hypotheses
Ref Expression
diafn.b 𝐡 = (Baseβ€˜πΎ)
diafn.l ≀ = (leβ€˜πΎ)
diafn.h 𝐻 = (LHypβ€˜πΎ)
diafn.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diafn ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š})
Distinct variable groups:   π‘₯, ≀   π‘₯,𝐡   π‘₯,𝐾   π‘₯,π‘Š
Allowed substitution hints:   𝐻(π‘₯)   𝐼(π‘₯)   𝑉(π‘₯)

Proof of Theorem diafn
Dummy variables 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6895 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
21rabex 5323 . . 3 {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑦} ∈ V
3 eqid 2724 . . 3 (𝑦 ∈ {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑦}) = (𝑦 ∈ {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑦})
42, 3fnmpti 6684 . 2 (𝑦 ∈ {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑦}) Fn {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š}
5 diafn.b . . . 4 𝐡 = (Baseβ€˜πΎ)
6 diafn.l . . . 4 ≀ = (leβ€˜πΎ)
7 diafn.h . . . 4 𝐻 = (LHypβ€˜πΎ)
8 eqid 2724 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
9 eqid 2724 . . . 4 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
10 diafn.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
115, 6, 7, 8, 9, 10diafval 40396 . . 3 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (𝑦 ∈ {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑦}))
1211fneq1d 6633 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝐼 Fn {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š} ↔ (𝑦 ∈ {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑦}) Fn {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š}))
134, 12mpbiri 258 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3424   class class class wbr 5139   ↦ cmpt 5222   Fn wfn 6529  β€˜cfv 6534  Basecbs 17145  lecple 17205  LHypclh 39349  LTrncltrn 39466  trLctrl 39523  DIsoAcdia 40393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-disoa 40394
This theorem is referenced by:  diadm  40400  diaelrnN  40410  diaf11N  40414
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