fsuppunbi - Metamath Proof Explorer

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Theorem fsuppunbi 9401

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 9375	. . . . 5 ⊢ Rel finSupp
2	1	brrelex12i 5709	. . . 4 ⊢ ((𝐹 ∪ 𝐺) finSupp 𝑍 → ((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ V))
3		unexb 7741	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ↔ (𝐹 ∪ 𝐺) ∈ V)
4		simpr 484	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin)
5	4	adantr 480	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin)
6		simprlr 779	. . . . . . . . . . . 12 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝐺 ∈ V)
7	6	suppun 8183	. . . . . . . . . . 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 9273	. . . . . . . . . 10 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐹 supp 𝑍) ∈ Fin)
9		fununfun 6584	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))
10	9	simpld 494	. . . . . . . . . . . . 13 ⊢ (Fun (𝐹 ∪ 𝐺) → Fun 𝐹)
11	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → Fun 𝐹)
12	11	adantr 480	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → Fun 𝐹)
13		simprll 778	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝐹 ∈ V)
14		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
15	14	adantl 481	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝑍 ∈ V)
16		funisfsupp 9379	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
17	12, 13, 15, 16	syl3anc 1373	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝐹 finSupp 𝑍)
19		uncom 4133	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
20	19	oveq1i 7415	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∪ 𝐺) supp 𝑍) = ((𝐺 ∪ 𝐹) supp 𝑍)
21	20	eleq1i 2825	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin ↔ ((𝐺 ∪ 𝐹) supp 𝑍) ∈ Fin)
22	21	biimpi 216	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin → ((𝐺 ∪ 𝐹) supp 𝑍) ∈ Fin)
23	22	adantl 481	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → ((𝐺 ∪ 𝐹) supp 𝑍) ∈ Fin)
24	23	adantr 480	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → ((𝐺 ∪ 𝐹) supp 𝑍) ∈ Fin)
25	13	suppun 8183	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐺 supp 𝑍) ⊆ ((𝐺 ∪ 𝐹) supp 𝑍))
26	24, 25	ssfid 9273	. . . . . . . . . 10 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐺 supp 𝑍) ∈ Fin)
27	9	simprd 495	. . . . . . . . . . . . 13 ⊢ (Fun (𝐹 ∪ 𝐺) → Fun 𝐺)
28	27	adantr 480	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → Fun 𝐺)
29	28	adantr 480	. . . . . . . . . . 11 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → Fun 𝐺)
30		funisfsupp 9379	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 𝑍 ∈ V) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
31	29, 6, 15, 30	syl3anc 1373	. . . . . . . . . 10 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin))
32	26, 31	mpbird 257	. . . . . . . . 9 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → 𝐺 finSupp 𝑍)
33	18, 32	jca 511	. . . . . . . 8 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍))
34	33	a1d 25	. . . . . . 7 ⊢ (((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) ∧ ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V)) → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))
35	34	ex 412	. . . . . 6 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) → (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍))))
36		fsuppimp 9380	. . . . . 6 ⊢ ((𝐹 ∪ 𝐺) finSupp 𝑍 → (Fun (𝐹 ∪ 𝐺) ∧ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin))
37	35, 36	syl11 33	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ 𝑍 ∈ V) → ((𝐹 ∪ 𝐺) finSupp 𝑍 → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍))))
38	3, 37	sylanbr 582	. . . 4 ⊢ (((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∪ 𝐺) finSupp 𝑍 → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍))))
39	2, 38	mpcom 38	. . 3 ⊢ ((𝐹 ∪ 𝐺) finSupp 𝑍 → (𝜑 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))
40	39	com12 32	. 2 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) finSupp 𝑍 → (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))
41		simpl 482	. . . . . 6 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → 𝐹 finSupp 𝑍)
42		simpr 484	. . . . . 6 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → 𝐺 finSupp 𝑍)
43	41, 42	fsuppun 9399	. . . . 5 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin)
44	43	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin)
45		fsuppunbi.u	. . . . . 6 ⊢ (𝜑 → Fun (𝐹 ∪ 𝐺))
46	45	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → Fun (𝐹 ∪ 𝐺))
47	1	brrelex1i 5710	. . . . . . 7 ⊢ (𝐹 finSupp 𝑍 → 𝐹 ∈ V)
48	1	brrelex1i 5710	. . . . . . 7 ⊢ (𝐺 finSupp 𝑍 → 𝐺 ∈ V)
49		unexg 7737	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∪ 𝐺) ∈ V)
50	47, 48, 49	syl2an 596	. . . . . 6 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → (𝐹 ∪ 𝐺) ∈ V)
51	50	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → (𝐹 ∪ 𝐺) ∈ V)
52	1	brrelex2i 5711	. . . . . . 7 ⊢ (𝐹 finSupp 𝑍 → 𝑍 ∈ V)
53	52	adantr 480	. . . . . 6 ⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → 𝑍 ∈ V)
54	53	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → 𝑍 ∈ V)
55		funisfsupp 9379	. . . . 5 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ (𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∪ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin))
56	46, 51, 54, 55	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → ((𝐹 ∪ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin))
57	44, 56	mpbird 257	. . 3 ⊢ ((𝜑 ∧ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)) → (𝐹 ∪ 𝐺) finSupp 𝑍)
58	57	ex 412	. 2 ⊢ (𝜑 → ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → (𝐹 ∪ 𝐺) finSupp 𝑍))
59	40, 58	impbid 212	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) finSupp 𝑍 ↔ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3459 ∪ cun 3924 class class class wbr 5119 Fun wfun 6525 (class class class)co 7405 supp csupp 8159 Fincfn 8959 finSupp cfsupp 9373

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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-supp 8160 df-1o 8480 df-en 8960 df-fin 8963 df-fsupp 9374

This theorem is referenced by: funsnfsupp 9404 lbsdiflsp0 33666

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