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

Theorem cantnfrescl 9620
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 482 . . . . . . 7 ((𝜑𝑛 ∈ (𝐷𝐵)) → ∅ ∈ 𝐴)
52, 4eqeltrd 2834 . . . . . 6 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋𝐴)
65ralrimiva 3140 . . . . 5 (𝜑 → ∀𝑛 ∈ (𝐷𝐵)𝑋𝐴)
71, 6raldifeq 4455 . . . 4 (𝜑 → (∀𝑛𝐵 𝑋𝐴 ↔ ∀𝑛𝐷 𝑋𝐴))
8 eqid 2733 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
98fmpt 7062 . . . 4 (∀𝑛𝐵 𝑋𝐴 ↔ (𝑛𝐵𝑋):𝐵𝐴)
10 eqid 2733 . . . . 5 (𝑛𝐷𝑋) = (𝑛𝐷𝑋)
1110fmpt 7062 . . . 4 (∀𝑛𝐷 𝑋𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴)
127, 9, 113bitr3g 313 . . 3 (𝜑 → ((𝑛𝐵𝑋):𝐵𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴))
13 cantnfs.b . . . . . 6 (𝜑𝐵 ∈ On)
1413mptexd 7178 . . . . 5 (𝜑 → (𝑛𝐵𝑋) ∈ V)
15 funmpt 6543 . . . . . 6 Fun (𝑛𝐵𝑋)
1615a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝐵𝑋))
17 cantnfrescl.d . . . . . . 7 (𝜑𝐷 ∈ On)
1817mptexd 7178 . . . . . 6 (𝜑 → (𝑛𝐷𝑋) ∈ V)
19 funmpt 6543 . . . . . 6 Fun (𝑛𝐷𝑋)
2018, 19jctir 522 . . . . 5 (𝜑 → ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋)))
2114, 16, 20jca31 516 . . . 4 (𝜑 → (((𝑛𝐵𝑋) ∈ V ∧ Fun (𝑛𝐵𝑋)) ∧ ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋))))
2217, 1, 2extmptsuppeq 8123 . . . 4 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
23 suppeqfsuppbi 9327 . . . 4 ((((𝑛𝐵𝑋) ∈ V ∧ Fun (𝑛𝐵𝑋)) ∧ ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋))) → (((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅) → ((𝑛𝐵𝑋) finSupp ∅ ↔ (𝑛𝐷𝑋) finSupp ∅)))
2421, 22, 23sylc 65 . . 3 (𝜑 → ((𝑛𝐵𝑋) finSupp ∅ ↔ (𝑛𝐷𝑋) finSupp ∅))
2512, 24anbi12d 632 . 2 (𝜑 → (((𝑛𝐵𝑋):𝐵𝐴 ∧ (𝑛𝐵𝑋) finSupp ∅) ↔ ((𝑛𝐷𝑋):𝐷𝐴 ∧ (𝑛𝐷𝑋) finSupp ∅)))
26 cantnfs.s . . 3 𝑆 = dom (𝐴 CNF 𝐵)
27 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
2826, 27, 13cantnfs 9610 . 2 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ ((𝑛𝐵𝑋):𝐵𝐴 ∧ (𝑛𝐵𝑋) finSupp ∅)))
29 cantnfrescl.t . . 3 𝑇 = dom (𝐴 CNF 𝐷)
3029, 27, 17cantnfs 9610 . 2 (𝜑 → ((𝑛𝐷𝑋) ∈ 𝑇 ↔ ((𝑛𝐷𝑋):𝐷𝐴 ∧ (𝑛𝐷𝑋) finSupp ∅)))
3125, 28, 303bitr4d 311 1 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3447  cdif 3911  wss 3914  c0 4286   class class class wbr 5109  cmpt 5192  dom cdm 5637  Oncon0 6321  Fun wfun 6494  wf 6496  (class class class)co 7361   supp csupp 8096   finSupp cfsupp 9311   CNF ccnf 9605
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-pow 5324  ax-pr 5388  ax-un 7676
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-pw 4566  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-pred 6257  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-ov 7364  df-oprab 7365  df-mpo 7366  df-supp 8097  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-seqom 8398  df-map 8773  df-fsupp 9312  df-cnf 9606
This theorem is referenced by:  cantnfres  9621
  Copyright terms: Public domain W3C validator