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Theorem suppeqfsuppbi 9419
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 9407 . . . . . 6 ((Fun 𝐹𝐹𝑈𝑍 ∈ V) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
51, 2, 3, 4syl3anc 1373 . . . . 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 9407 . . . . . . . . 9 ((Fun 𝐺𝐺𝑉𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
138, 10, 11, 12syl3anc 1373 . . . . . . . 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 2829 . . . . . 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 9403 . . . . 5 Rel finSupp
2322brrelex2i 5742 . . . 4 (𝐹 finSupp 𝑍𝑍 ∈ V)
2422brrelex2i 5742 . . . 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 1540  wcel 2108  Vcvv 3480   class class class wbr 5143  Fun wfun 6555  (class class class)co 7431   supp csupp 8185  Fincfn 8985   finSupp cfsupp 9401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-fsupp 9402
This theorem is referenced by:  cantnfrescl  9716
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