Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnfrescl Structured version   Visualization version   GIF version

Theorem cantnfrescl 8736
 Description: A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
cantnfrescl.d (𝜑𝐷 ∈ On)
cantnfrescl.b (𝜑𝐵𝐷)
cantnfrescl.x ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
cantnfrescl.a (𝜑 → ∅ ∈ 𝐴)
cantnfrescl.t 𝑇 = dom (𝐴 CNF 𝐷)
Assertion
Ref Expression
cantnfrescl (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
Distinct variable groups:   𝐵,𝑛   𝐷,𝑛   𝐴,𝑛   𝜑,𝑛
Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑋(𝑛)

Proof of Theorem cantnfrescl
StepHypRef Expression
1 cantnfrescl.b . . . . 5 (𝜑𝐵𝐷)
2 cantnfrescl.x . . . . . . 7 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
3 cantnfrescl.a . . . . . . . 8 (𝜑 → ∅ ∈ 𝐴)
43adantr 466 . . . . . . 7 ((𝜑𝑛 ∈ (𝐷𝐵)) → ∅ ∈ 𝐴)
52, 4eqeltrd 2849 . . . . . 6 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋𝐴)
65ralrimiva 3114 . . . . 5 (𝜑 → ∀𝑛 ∈ (𝐷𝐵)𝑋𝐴)
71, 6raldifeq 4198 . . . 4 (𝜑 → (∀𝑛𝐵 𝑋𝐴 ↔ ∀𝑛𝐷 𝑋𝐴))
8 eqid 2770 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
98fmpt 6523 . . . 4 (∀𝑛𝐵 𝑋𝐴 ↔ (𝑛𝐵𝑋):𝐵𝐴)
10 eqid 2770 . . . . 5 (𝑛𝐷𝑋) = (𝑛𝐷𝑋)
1110fmpt 6523 . . . 4 (∀𝑛𝐷 𝑋𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴)
127, 9, 113bitr3g 302 . . 3 (𝜑 → ((𝑛𝐵𝑋):𝐵𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴))
13 cantnfs.b . . . . . 6 (𝜑𝐵 ∈ On)
14 mptexg 6627 . . . . . 6 (𝐵 ∈ On → (𝑛𝐵𝑋) ∈ V)
1513, 14syl 17 . . . . 5 (𝜑 → (𝑛𝐵𝑋) ∈ V)
16 funmpt 6069 . . . . . 6 Fun (𝑛𝐵𝑋)
1716a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝐵𝑋))
18 cantnfrescl.d . . . . . . 7 (𝜑𝐷 ∈ On)
19 mptexg 6627 . . . . . . 7 (𝐷 ∈ On → (𝑛𝐷𝑋) ∈ V)
2018, 19syl 17 . . . . . 6 (𝜑 → (𝑛𝐷𝑋) ∈ V)
21 funmpt 6069 . . . . . 6 Fun (𝑛𝐷𝑋)
2220, 21jctir 504 . . . . 5 (𝜑 → ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋)))
2315, 17, 22jca31 498 . . . 4 (𝜑 → (((𝑛𝐵𝑋) ∈ V ∧ Fun (𝑛𝐵𝑋)) ∧ ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋))))
2418, 1, 2extmptsuppeq 7469 . . . 4 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
25 suppeqfsuppbi 8444 . . . 4 ((((𝑛𝐵𝑋) ∈ V ∧ Fun (𝑛𝐵𝑋)) ∧ ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋))) → (((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅) → ((𝑛𝐵𝑋) finSupp ∅ ↔ (𝑛𝐷𝑋) finSupp ∅)))
2623, 24, 25sylc 65 . . 3 (𝜑 → ((𝑛𝐵𝑋) finSupp ∅ ↔ (𝑛𝐷𝑋) finSupp ∅))
2712, 26anbi12d 608 . 2 (𝜑 → (((𝑛𝐵𝑋):𝐵𝐴 ∧ (𝑛𝐵𝑋) finSupp ∅) ↔ ((𝑛𝐷𝑋):𝐷𝐴 ∧ (𝑛𝐷𝑋) finSupp ∅)))
28 cantnfs.s . . 3 𝑆 = dom (𝐴 CNF 𝐵)
29 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
3028, 29, 13cantnfs 8726 . 2 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ ((𝑛𝐵𝑋):𝐵𝐴 ∧ (𝑛𝐵𝑋) finSupp ∅)))
31 cantnfrescl.t . . 3 𝑇 = dom (𝐴 CNF 𝐷)
3231, 29, 18cantnfs 8726 . 2 (𝜑 → ((𝑛𝐷𝑋) ∈ 𝑇 ↔ ((𝑛𝐷𝑋):𝐷𝐴 ∧ (𝑛𝐷𝑋) finSupp ∅)))
3327, 30, 323bitr4d 300 1 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144  ∀wral 3060  Vcvv 3349   ∖ cdif 3718   ⊆ wss 3721  ∅c0 4061   class class class wbr 4784   ↦ cmpt 4861  dom cdm 5249  Oncon0 5866  Fun wfun 6025  ⟶wf 6027  (class class class)co 6792   supp csupp 7445   finSupp cfsupp 8430   CNF ccnf 8721 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-seqom 7695  df-map 8010  df-fsupp 8431  df-cnf 8722 This theorem is referenced by:  cantnfres  8737
 Copyright terms: Public domain W3C validator