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	⊢ (((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍 ↔ 𝐺 finSupp 𝑍)))

Proof of Theorem suppeqfsuppbi

Step	Hyp	Ref	Expression
1		simprlr 780	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ ((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺))) → Fun 𝐹)
2		simprll 779	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ ((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺))) → 𝐹 ∈ 𝑈)
3		simpl 482	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ ((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺))) → 𝑍 ∈ V)
4		funisfsupp 9407	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑈 ∧ 𝑍 ∈ V) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
5	1, 2, 3, 4	syl3anc 1373	. . . . 5 ⊢ ((𝑍 ∈ V ∧ ((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺))) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
6	5	adantr 480	. . . 4 ⊢ (((𝑍 ∈ V ∧ ((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
7		simpr 484	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ Fun 𝐺) → Fun 𝐺)
8	7	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → Fun 𝐺)
9		simpl 482	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ Fun 𝐺) → 𝐺 ∈ 𝑉)
10	9	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝐺 ∈ 𝑉)
11		simpr 484	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
12		funisfsupp 9407	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐺 ∈ 𝑉 ∧ 𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
13	8, 10, 11, 12	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ 𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
14	13	ex 412	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ Fun 𝐺) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
15	14	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺)) → (𝑍 ∈ V → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)))
16	15	impcom 407	. . . . 5 ⊢ ((𝑍 ∈ V ∧ ((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ 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))
18	17	bicomd 223	. . . . 5 ⊢ ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → ((𝐺 supp 𝑍) ∈ Fin ↔ (𝐹 supp 𝑍) ∈ Fin))
19	16, 18	sylan9bb 509	. . . 4 ⊢ (((𝑍 ∈ V ∧ ((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐺 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
20	6, 19	bitr4d 282	. . 3 ⊢ (((𝑍 ∈ V ∧ ((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺))) ∧ (𝐹 supp 𝑍) = (𝐺 supp 𝑍)) → (𝐹 finSupp 𝑍 ↔ 𝐺 finSupp 𝑍))
21	20	exp31 419	. 2 ⊢ (𝑍 ∈ V → (((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍 ↔ 𝐺 finSupp 𝑍))))
22		relfsupp 9403	. . . . 5 ⊢ Rel finSupp
23	22	brrelex2i 5742	. . . 4 ⊢ (𝐹 finSupp 𝑍 → 𝑍 ∈ V)
24	22	brrelex2i 5742	. . . 4 ⊢ (𝐺 finSupp 𝑍 → 𝑍 ∈ V)
25	23, 24	pm5.21ni 377	. . 3 ⊢ (¬ 𝑍 ∈ V → (𝐹 finSupp 𝑍 ↔ 𝐺 finSupp 𝑍))
26	25	2a1d 26	. 2 ⊢ (¬ 𝑍 ∈ V → (((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍 ↔ 𝐺 finSupp 𝑍))))
27	21, 26	pm2.61i 182	1 ⊢ (((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍 ↔ 𝐺 finSupp 𝑍)))

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