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Theorem suppeqfsuppbi 8405
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 822 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → Fun 𝐹)
2 simprll 821 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → 𝐹𝑈)
3 simpl 474 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → 𝑍 ∈ V)
4 funisfsupp 8396 . . . . . 6 ((Fun 𝐹𝐹𝑈𝑍 ∈ V) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
51, 2, 3, 4syl3anc 1439 . . . . 5 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
65adantr 472 . . . 4 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
7 simpr 479 . . . . . . . . . 10 ((𝐺𝑉 ∧ Fun 𝐺) → Fun 𝐺)
87adantr 472 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → Fun 𝐺)
9 simpl 474 . . . . . . . . . 10 ((𝐺𝑉 ∧ Fun 𝐺) → 𝐺𝑉)
109adantr 472 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝐺𝑉)
11 simpr 479 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
12 funisfsupp 8396 . . . . . . . . 9 ((Fun 𝐺𝐺𝑉𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
138, 10, 11, 12syl3anc 1439 . . . . . . . 8 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
1413ex 449 . . . . . . 7 ((𝐺𝑉 ∧ Fun 𝐺) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
1514adantl 473 . . . . . 6 (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
1615impcom 445 . . . . 5 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
17 eleq1 2791 . . . . . 6 ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → ((𝐹 supp 𝑍) ∈ Fin ↔ (𝐺 supp 𝑍) ∈ Fin))
1817bicomd 213 . . . . 5 ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → ((𝐺 supp 𝑍) ∈ Fin ↔ (𝐹 supp 𝑍) ∈ Fin))
1916, 18sylan9bb 738 . . . 4 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐺 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
206, 19bitr4d 271 . . 3 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))
2120exp31 631 . 2 (𝑍 ∈ V → (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))))
22 relfsupp 8393 . . . . 5 Rel finSupp
2322brrelex2i 5268 . . . 4 (𝐹 finSupp 𝑍𝑍 ∈ V)
2422brrelex2i 5268 . . . 4 (𝐺 finSupp 𝑍𝑍 ∈ V)
2523, 24pm5.21ni 366 . . 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 176 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 196  wa 383   = wceq 1596  wcel 2103  Vcvv 3304   class class class wbr 4760  Fun wfun 5995  (class class class)co 6765   supp csupp 7415  Fincfn 8072   finSupp cfsupp 8391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-fsupp 8392
This theorem is referenced by:  cantnfrescl  8686
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