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Theorem cantnfrescl 9130
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 484 . . . . . . 7 ((𝜑𝑛 ∈ (𝐷𝐵)) → ∅ ∈ 𝐴)
52, 4eqeltrd 2916 . . . . . 6 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋𝐴)
65ralrimiva 3177 . . . . 5 (𝜑 → ∀𝑛 ∈ (𝐷𝐵)𝑋𝐴)
71, 6raldifeq 4421 . . . 4 (𝜑 → (∀𝑛𝐵 𝑋𝐴 ↔ ∀𝑛𝐷 𝑋𝐴))
8 eqid 2824 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
98fmpt 6862 . . . 4 (∀𝑛𝐵 𝑋𝐴 ↔ (𝑛𝐵𝑋):𝐵𝐴)
10 eqid 2824 . . . . 5 (𝑛𝐷𝑋) = (𝑛𝐷𝑋)
1110fmpt 6862 . . . 4 (∀𝑛𝐷 𝑋𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴)
127, 9, 113bitr3g 316 . . 3 (𝜑 → ((𝑛𝐵𝑋):𝐵𝐴 ↔ (𝑛𝐷𝑋):𝐷𝐴))
13 cantnfs.b . . . . . 6 (𝜑𝐵 ∈ On)
1413mptexd 6975 . . . . 5 (𝜑 → (𝑛𝐵𝑋) ∈ V)
15 funmpt 6381 . . . . . 6 Fun (𝑛𝐵𝑋)
1615a1i 11 . . . . 5 (𝜑 → Fun (𝑛𝐵𝑋))
17 cantnfrescl.d . . . . . . 7 (𝜑𝐷 ∈ On)
1817mptexd 6975 . . . . . 6 (𝜑 → (𝑛𝐷𝑋) ∈ V)
19 funmpt 6381 . . . . . 6 Fun (𝑛𝐷𝑋)
2018, 19jctir 524 . . . . 5 (𝜑 → ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋)))
2114, 16, 20jca31 518 . . . 4 (𝜑 → (((𝑛𝐵𝑋) ∈ V ∧ Fun (𝑛𝐵𝑋)) ∧ ((𝑛𝐷𝑋) ∈ V ∧ Fun (𝑛𝐷𝑋))))
2217, 1, 2extmptsuppeq 7844 . . . 4 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
23 suppeqfsuppbi 8838 . . . 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 633 . 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 9120 . 2 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ ((𝑛𝐵𝑋):𝐵𝐴 ∧ (𝑛𝐵𝑋) finSupp ∅)))
29 cantnfrescl.t . . 3 𝑇 = dom (𝐴 CNF 𝐷)
3029, 27, 17cantnfs 9120 . 2 (𝜑 → ((𝑛𝐷𝑋) ∈ 𝑇 ↔ ((𝑛𝐷𝑋):𝐷𝐴 ∧ (𝑛𝐷𝑋) finSupp ∅)))
3125, 28, 303bitr4d 314 1 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  cdif 3916  wss 3919  c0 4275   class class class wbr 5052  cmpt 5132  dom cdm 5542  Oncon0 6178  Fun wfun 6337  wf 6339  (class class class)co 7145   supp csupp 7820   finSupp cfsupp 8824   CNF ccnf 9115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-supp 7821  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-seqom 8074  df-map 8398  df-fsupp 8825  df-cnf 9116
This theorem is referenced by:  cantnfres  9131
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