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Theorem suppeqfsuppbi 9417
Description: If two functions have the same support, one function is finitely supported iff the other one is finitely supported. (Contributed by AV, 30-Jun-2019.)
Assertion
Ref Expression
suppeqfsuppbi (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍)))

Proof of Theorem suppeqfsuppbi
StepHypRef Expression
1 simprlr 780 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → Fun 𝐹)
2 simprll 779 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → 𝐹𝑈)
3 simpl 482 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → 𝑍 ∈ V)
4 funisfsupp 9405 . . . . . 6 ((Fun 𝐹𝐹𝑈𝑍 ∈ V) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
51, 2, 3, 4syl3anc 1370 . . . . 5 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
65adantr 480 . . . 4 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
7 simpr 484 . . . . . . . . . 10 ((𝐺𝑉 ∧ Fun 𝐺) → Fun 𝐺)
87adantr 480 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → Fun 𝐺)
9 simpl 482 . . . . . . . . . 10 ((𝐺𝑉 ∧ Fun 𝐺) → 𝐺𝑉)
109adantr 480 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝐺𝑉)
11 simpr 484 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
12 funisfsupp 9405 . . . . . . . . 9 ((Fun 𝐺𝐺𝑉𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
138, 10, 11, 12syl3anc 1370 . . . . . . . 8 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
1413ex 412 . . . . . . 7 ((𝐺𝑉 ∧ Fun 𝐺) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
1514adantl 481 . . . . . 6 (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
1615impcom 407 . . . . 5 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
17 eleq1 2827 . . . . . 6 ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → ((𝐹 supp 𝑍) ∈ Fin ↔ (𝐺 supp 𝑍) ∈ Fin))
1817bicomd 223 . . . . 5 ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → ((𝐺 supp 𝑍) ∈ Fin ↔ (𝐹 supp 𝑍) ∈ Fin))
1916, 18sylan9bb 509 . . . 4 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐺 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
206, 19bitr4d 282 . . 3 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))
2120exp31 419 . 2 (𝑍 ∈ V → (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))))
22 relfsupp 9401 . . . . 5 Rel finSupp
2322brrelex2i 5746 . . . 4 (𝐹 finSupp 𝑍𝑍 ∈ V)
2422brrelex2i 5746 . . . 4 (𝐺 finSupp 𝑍𝑍 ∈ V)
2523, 24pm5.21ni 377 . . 3 𝑍 ∈ V → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))
26252a1d 26 . 2 𝑍 ∈ V → (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))))
2721, 26pm2.61i 182 1 (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478   class class class wbr 5148  Fun wfun 6557  (class class class)co 7431   supp csupp 8184  Fincfn 8984   finSupp cfsupp 9399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-fsupp 9400
This theorem is referenced by:  cantnfrescl  9714
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