fsuppunbi - Metamath Proof Explorer

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Theorem fsuppunbi 9292

Description: If the union of two classes/functions is a function, this union is finitely supported iff the two functions are finitely supported. (Contributed by AV, 18-Jun-2019.)

**Hypothesis**
Ref	Expression
fsuppunbi.u	⊢ (𝜑 → Fun (𝐹 ∪ 𝐺))

**Assertion**
Ref	Expression
fsuppunbi	⊢ (𝜑 → ((𝐹 ∪ 𝐺) finSupp 𝑍 ↔ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))

**Proof of Theorem fsuppunbi**
Step	Hyp	Ref	Expression
1		relfsupp 9266	. . . . 5 ⊢ Rel finSupp
2	1	brrelex12i 5673	. . . 4 ⊢ ((𝐹 ∪ 𝐺) finSupp 𝑍 → ((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ V))
3		unexb 7690	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ↔ (𝐹 ∪ 𝐺) ∈ V)
4		simpr 485	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin)
5	4	adantr 481	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin)
6		simprlr 785	. . . . . . . . . . . 12 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝐺 ∈ V)
7	6	suppun 8124	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))
8	5, 7	ssfid 9169	. . . . . . . . . 10 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐹 supp 𝑍) ∈ Fin)
9		fununfun 6533	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))
10	9	simpld 495	. . . . . . . . . . . . 13 ⊢ (Fun (𝐹 ∪ 𝐺) → Fun 𝐹)
11	10	adantr 481	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → Fun 𝐹)
12	11	adantr 481	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → Fun 𝐹)
13		simprll 784	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝐹 ∈ V)
14		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
15	14	adantl 482	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝑍 ∈ V)
16		funisfsupp 9270	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
17	12, 13, 15, 16	syl3anc 1379	. . . . . . . . . 10 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
18	8, 17	mpbird 258	. . . . . . . . 9 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝐹 finSupp 𝑍)
19		uncom 4088	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
20	19	oveq1i 7366	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∪ 𝐺) supp 𝑍) = ((𝐺 ∪ 𝐹) supp 𝑍)
21	20	eleq1i 2830	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin ↔ ((𝐺 ∪ 𝐹) supp 𝑍) ∈ Fin)
22	21	bilani 505	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → ((𝐺 ∪ 𝐹) supp 𝑍) ∈ Fin)
23	22	adantr 481	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → ((𝐺 ∪ 𝐹) supp 𝑍) ∈ Fin)
24	13	suppun 8124	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐺 supp 𝑍) ⊆ ((𝐺 ∪ 𝐹) supp 𝑍))
25	23, 24	ssfid 9169	. . . . . . . . . 10 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐺 supp 𝑍) ∈ Fin)
26	9	simprd 496	. . . . . . . . . . . . 13 ⊢ (Fun (𝐹 ∪ 𝐺) → Fun 𝐺)
27	26	adantr 481	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → Fun 𝐺)
28	27	adantr 481	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → Fun 𝐺)
29		funisfsupp 9270	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
30	28, 6, 15, 29	syl3anc 1379	. . . . . . . . . 10 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
31	25, 30	mpbird 258	. . . . . . . . 9 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝐺 finSupp 𝑍)
32	18, 31	jca 516	. . . . . . . 8 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍))
33	32	a1d 25	. . . . . . 7 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))
34	33	ex 413	. . . . . 6 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍))))
35		fsuppimp 9271	. . . . . 6 ⊢ ((𝐹 ∪ 𝐺) finSupp 𝑍 → (Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin))
36	34, 35	syl11 33	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V) → ((𝐹 ∪ 𝐺) finSupp 𝑍 → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍))))
37	3, 36	sylanbr 588	. . . 4 ⊢ (((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∪ 𝐺) finSupp 𝑍 → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍))))
38	2, 37	mpcom 38	. . 3 ⊢ ((𝐹 ∪ 𝐺) finSupp 𝑍 → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))
39	38	com12 32	. 2 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) finSupp 𝑍 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))
40		simpl 483	. . . . . 6 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → 𝐹 finSupp 𝑍)
41		simpr 485	. . . . . 6 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → 𝐺 finSupp 𝑍)
42	40, 41	fsuppun 9290	. . . . 5 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin)
43	42	adantl 482	. . . 4 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin)
44		fsuppunbi.u	. . . . . 6 ⊢ (𝜑 → Fun (𝐹 ∪ 𝐺))
45	44	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → Fun (𝐹 ∪ 𝐺))
46	1	brrelex1i 5674	. . . . . . 7 ⊢ (𝐹 finSupp 𝑍 → 𝐹 ∈ V)
47	1	brrelex1i 5674	. . . . . . 7 ⊢ (𝐺 finSupp 𝑍 → 𝐺 ∈ V)
48		unexg 7686	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∪ 𝐺) ∈ V)
49	46, 47, 48	syl2an 602	. . . . . 6 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → (𝐹 ∪ 𝐺) ∈ V)
50	49	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → (𝐹 ∪ 𝐺) ∈ V)
51	1	brrelex2i 5675	. . . . . . 7 ⊢ (𝐹 finSupp 𝑍 → 𝑍 ∈ V)
52	51	adantr 481	. . . . . 6 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → 𝑍 ∈ V)
53	52	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → 𝑍 ∈ V)
54		funisfsupp 9270	. . . . 5 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ (𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∪ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin))
55	45, 50, 53, 54	syl3anc 1379	. . . 4 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → ((𝐹 ∪ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin))
56	43, 55	mpbird 258	. . 3 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → (𝐹 ∪ 𝐺) finSupp 𝑍)
57	56	ex 413	. 2 ⊢ (𝜑 → ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → (𝐹 ∪ 𝐺) finSupp 𝑍))
58	39, 57	impbid 213	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) finSupp 𝑍 ↔ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2119 Vcvv 3431 ∪ cun 3881 class class class wbr 5072 Fun wfun 6479 (class class class)co 7356 supp csupp 8100 Fincfn 8883 finSupp cfsupp 9264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-supp 8101 df-1o 8395 df-en 8884 df-fin 8887 df-fsupp 9265

This theorem is referenced by: funsnfsupp 9295 lbsdiflsp0 33810

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