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Theorem diafn 39547
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 6859 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
21rabex 5293 . . 3 {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑦} ∈ V
3 eqid 2733 . . 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 6648 . 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 2733 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
9 eqid 2733 . . . 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 39544 . . 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 6599 . 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 397   = wceq 1542   ∈ wcel 2107  {crab 3406   class class class wbr 5109   ↦ cmpt 5192   Fn wfn 6495  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:  diadm  39548  diaelrnN  39558  diaf11N  39562
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