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Theorem suppeqfsuppbi 8839
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 778 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → Fun 𝐹)
2 simprll 777 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → 𝐹𝑈)
3 simpl 485 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → 𝑍 ∈ V)
4 funisfsupp 8830 . . . . . 6 ((Fun 𝐹𝐹𝑈𝑍 ∈ V) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
51, 2, 3, 4syl3anc 1365 . . . . 5 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
65adantr 483 . . . 4 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
7 simpr 487 . . . . . . . . . 10 ((𝐺𝑉 ∧ Fun 𝐺) → Fun 𝐺)
87adantr 483 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → Fun 𝐺)
9 simpl 485 . . . . . . . . . 10 ((𝐺𝑉 ∧ Fun 𝐺) → 𝐺𝑉)
109adantr 483 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝐺𝑉)
11 simpr 487 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
12 funisfsupp 8830 . . . . . . . . 9 ((Fun 𝐺𝐺𝑉𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
138, 10, 11, 12syl3anc 1365 . . . . . . . 8 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
1413ex 415 . . . . . . 7 ((𝐺𝑉 ∧ Fun 𝐺) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
1514adantl 484 . . . . . 6 (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
1615impcom 410 . . . . 5 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
17 eleq1 2898 . . . . . 6 ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → ((𝐹 supp 𝑍) ∈ Fin ↔ (𝐺 supp 𝑍) ∈ Fin))
1817bicomd 225 . . . . 5 ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → ((𝐺 supp 𝑍) ∈ Fin ↔ (𝐹 supp 𝑍) ∈ Fin))
1916, 18sylan9bb 512 . . . 4 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐺 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
206, 19bitr4d 284 . . 3 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))
2120exp31 422 . 2 (𝑍 ∈ V → (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))))
22 relfsupp 8827 . . . . 5 Rel finSupp
2322brrelex2i 5602 . . . 4 (𝐹 finSupp 𝑍𝑍 ∈ V)
2422brrelex2i 5602 . . . 4 (𝐺 finSupp 𝑍𝑍 ∈ V)
2523, 24pm5.21ni 381 . . 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 184 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 208  wa 398   = wceq 1530  wcel 2107  Vcvv 3493   class class class wbr 5057  Fun wfun 6342  (class class class)co 7148   supp csupp 7822  Fincfn 8501   finSupp cfsupp 8825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-fsupp 8826
This theorem is referenced by:  cantnfrescl  9131
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