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Theorem cantnfrescl 9127
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 481 . . . . . . 7 ((𝜑𝑛 ∈ (𝐷𝐵)) → ∅ ∈ 𝐴)
52, 4eqeltrd 2910 . . . . . 6 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋𝐴)
65ralrimiva 3179 . . . . 5 (𝜑 → ∀𝑛 ∈ (𝐷𝐵)𝑋𝐴)
71, 6raldifeq 4435 . . . 4 (𝜑 → (∀𝑛𝐵 𝑋𝐴 ↔ ∀𝑛𝐷 𝑋𝐴))
8 eqid 2818 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
98fmpt 6866 . . . 4 (∀𝑛𝐵 𝑋𝐴 ↔ (𝑛𝐵𝑋):𝐵𝐴)
10 eqid 2818 . . . . 5 (𝑛𝐷𝑋) = (𝑛𝐷𝑋)
1110fmpt 6866 . . . 4 (∀𝑛𝐷 𝑋𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴)
127, 9, 113bitr3g 314 . . 3 (𝜑 → ((𝑛𝐵𝑋):𝐵𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴))
13 cantnfs.b . . . . . 6 (𝜑𝐵 ∈ On)
1413mptexd 6978 . . . . 5 (𝜑 → (𝑛𝐵𝑋) ∈ V)
15 funmpt 6386 . . . . . 6 Fun (𝑛𝐵𝑋)
1615a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝐵𝑋))
17 cantnfrescl.d . . . . . . 7 (𝜑𝐷 ∈ On)
1817mptexd 6978 . . . . . 6 (𝜑 → (𝑛𝐷𝑋) ∈ V)
19 funmpt 6386 . . . . . 6 Fun (𝑛𝐷𝑋)
2018, 19jctir 521 . . . . 5 (𝜑 → ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋)))
2114, 16, 20jca31 515 . . . 4 (𝜑 → (((𝑛𝐵𝑋) ∈ V ∧ Fun (𝑛𝐵𝑋)) ∧ ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋))))
2217, 1, 2extmptsuppeq 7843 . . . 4 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
23 suppeqfsuppbi 8835 . . . 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 630 . 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 9117 . 2 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ ((𝑛𝐵𝑋):𝐵𝐴 ∧ (𝑛𝐵𝑋) finSupp ∅)))
29 cantnfrescl.t . . 3 𝑇 = dom (𝐴 CNF 𝐷)
3029, 27, 17cantnfs 9117 . 2 (𝜑 → ((𝑛𝐷𝑋) ∈ 𝑇 ↔ ((𝑛𝐷𝑋):𝐷𝐴 ∧ (𝑛𝐷𝑋) finSupp ∅)))
3125, 28, 303bitr4d 312 1 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  wss 3933  c0 4288   class class class wbr 5057  cmpt 5137  dom cdm 5548  Oncon0 6184  Fun wfun 6342  wf 6344  (class class class)co 7145   supp csupp 7819   finSupp cfsupp 8821   CNF ccnf 9112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seqom 8073  df-map 8397  df-fsupp 8822  df-cnf 9113
This theorem is referenced by:  cantnfres  9128
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