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Theorem cantnfrescl 8611
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 480 . . . . . . 7 ((𝜑𝑛 ∈ (𝐷𝐵)) → ∅ ∈ 𝐴)
52, 4eqeltrd 2730 . . . . . 6 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋𝐴)
65ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑛 ∈ (𝐷𝐵)𝑋𝐴)
71, 6raldifeq 4092 . . . 4 (𝜑 → (∀𝑛𝐵 𝑋𝐴 ↔ ∀𝑛𝐷 𝑋𝐴))
8 eqid 2651 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
98fmpt 6421 . . . 4 (∀𝑛𝐵 𝑋𝐴 ↔ (𝑛𝐵𝑋):𝐵𝐴)
10 eqid 2651 . . . . 5 (𝑛𝐷𝑋) = (𝑛𝐷𝑋)
1110fmpt 6421 . . . 4 (∀𝑛𝐷 𝑋𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴)
127, 9, 113bitr3g 302 . . 3 (𝜑 → ((𝑛𝐵𝑋):𝐵𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴))
13 cantnfs.b . . . . . 6 (𝜑𝐵 ∈ On)
14 mptexg 6525 . . . . . 6 (𝐵 ∈ On → (𝑛𝐵𝑋) ∈ V)
1513, 14syl 17 . . . . 5 (𝜑 → (𝑛𝐵𝑋) ∈ V)
16 funmpt 5964 . . . . . 6 Fun (𝑛𝐵𝑋)
1716a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝐵𝑋))
18 cantnfrescl.d . . . . . . 7 (𝜑𝐷 ∈ On)
19 mptexg 6525 . . . . . . 7 (𝐷 ∈ On → (𝑛𝐷𝑋) ∈ V)
2018, 19syl 17 . . . . . 6 (𝜑 → (𝑛𝐷𝑋) ∈ V)
21 funmpt 5964 . . . . . 6 Fun (𝑛𝐷𝑋)
2220, 21jctir 560 . . . . 5 (𝜑 → ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋)))
2315, 17, 22jca31 556 . . . 4 (𝜑 → (((𝑛𝐵𝑋) ∈ V ∧ Fun (𝑛𝐵𝑋)) ∧ ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋))))
2418, 1, 2extmptsuppeq 7364 . . . 4 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
25 suppeqfsuppbi 8330 . . . 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 747 . 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 8601 . 2 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ ((𝑛𝐵𝑋):𝐵𝐴 ∧ (𝑛𝐵𝑋) finSupp ∅)))
31 cantnfrescl.t . . 3 𝑇 = dom (𝐴 CNF 𝐷)
3231, 29, 18cantnfs 8601 . 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 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  wss 3607  c0 3948   class class class wbr 4685  cmpt 4762  dom cdm 5143  Oncon0 5761  Fun wfun 5920  wf 5922  (class class class)co 6690   supp csupp 7340   finSupp cfsupp 8316   CNF ccnf 8596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-map 7901  df-fsupp 8317  df-cnf 8597
This theorem is referenced by:  cantnfres  8612
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