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Theorem suppeqfsuppbi 8240
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 802 . . . . . . 7 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → Fun 𝐹)
2 simprll 801 . . . . . . 7 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → 𝐹𝑈)
3 simpl 473 . . . . . . 7 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → 𝑍 ∈ V)
4 funisfsupp 8231 . . . . . . 7 ((Fun 𝐹𝐹𝑈𝑍 ∈ V) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
51, 2, 3, 4syl3anc 1323 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
65adantr 481 . . . . 5 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
7 simpr 477 . . . . . . . . . . 11 ((𝐺𝑉 ∧ Fun 𝐺) → Fun 𝐺)
87adantr 481 . . . . . . . . . 10 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → Fun 𝐺)
9 simpl 473 . . . . . . . . . . 11 ((𝐺𝑉 ∧ Fun 𝐺) → 𝐺𝑉)
109adantr 481 . . . . . . . . . 10 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝐺𝑉)
11 simpr 477 . . . . . . . . . 10 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
12 funisfsupp 8231 . . . . . . . . . 10 ((Fun 𝐺𝐺𝑉𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
138, 10, 11, 12syl3anc 1323 . . . . . . . . 9 (((𝐺𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
1413ex 450 . . . . . . . 8 ((𝐺𝑉 ∧ Fun 𝐺) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
1514adantl 482 . . . . . . 7 (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
1615impcom 446 . . . . . 6 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
17 eleq1 2686 . . . . . . 7 ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → ((𝐹 supp 𝑍) ∈ Fin ↔ (𝐺 supp 𝑍) ∈ Fin))
1817bicomd 213 . . . . . 6 ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → ((𝐺 supp 𝑍) ∈ Fin ↔ (𝐹 supp 𝑍) ∈ Fin))
1916, 18sylan9bb 735 . . . . 5 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐺 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
206, 19bitr4d 271 . . . 4 (((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))
2120ex 450 . . 3 ((𝑍 ∈ V ∧ ((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺))) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍)))
2221ex 450 . 2 (𝑍 ∈ V → (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))))
23 relfsupp 8228 . . . . 5 Rel finSupp
2423brrelex2i 5124 . . . 4 (𝐹 finSupp 𝑍𝑍 ∈ V)
2523brrelex2i 5124 . . . 4 (𝐺 finSupp 𝑍𝑍 ∈ V)
2624, 25pm5.21ni 367 . . 3 𝑍 ∈ V → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))
27262a1d 26 . 2 𝑍 ∈ V → (((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍))))
2822, 27pm2.61i 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 384   = wceq 1480  wcel 1987  Vcvv 3189   class class class wbr 4618  Fun wfun 5846  (class class class)co 6610   supp csupp 7247  Fincfn 7906   finSupp cfsupp 8226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-fsupp 8227
This theorem is referenced by:  cantnfrescl  8524
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