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Theorem diafn 40501
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 6904 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
21rabex 5328 . . 3 {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑦} ∈ V
3 eqid 2728 . . 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 6692 . 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 2728 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
9 eqid 2728 . . . 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 40498 . . 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 6641 . 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 1534   ∈ wcel 2099  {crab 3428   class class class wbr 5142   ↦ cmpt 5225   Fn wfn 6537  β€˜cfv 6542  Basecbs 17173  lecple 17233  LHypclh 39451  LTrncltrn 39568  trLctrl 39625  DIsoAcdia 40495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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-disoa 40496
This theorem is referenced by:  diadm  40502  diaelrnN  40512  diaf11N  40516
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