suppovss - Mathbox for Thierry Arnoux

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Theorem suppovss 32773

Description: A bound for the support of an operation. (Contributed by Thierry Arnoux, 19-Jul-2023.)

**Hypotheses**
Ref	Expression
suppovss.f	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
suppovss.g	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ 𝐶))
suppovss.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
suppovss.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
suppovss.z	⊢ (𝜑 → 𝑍 ∈ 𝐷)
suppovss.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)

**Assertion**
Ref	Expression
suppovss	⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))

Distinct variable groups: 𝐴,𝑘,𝑥,𝑦 𝐵,𝑘,𝑥,𝑦 𝑥,𝐷,𝑦 𝑥,𝐹,𝑦 𝑘,𝐺,𝑥,𝑦 𝑘,𝑍,𝑥,𝑦 𝜑,𝑘,𝑥,𝑦 Allowed substitution hints: 𝐶(𝑥,𝑦,𝑘) 𝐷(𝑘) 𝐹(𝑘) 𝑉(𝑥,𝑦,𝑘) 𝑊(𝑥,𝑦,𝑘)

Proof of Theorem suppovss Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppovss.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)
2	1	ralrimivva 3182	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
3		suppovss.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	fmpo 8010	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷)
5	2, 4	sylib 219	. 2 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)⟶𝐷)
6		simpr 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
7	6	fveq2d 6831	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
8		df-ov 7359	. . . . . . . 8 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
9		simpllr 781	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
10	9	eldifad 3895	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ 𝐴)
11		simplr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦 ∈ 𝐵)
12		simplll 780	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝜑)
13	12, 10, 11, 1	syl12anc 842	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 ∈ 𝐷)
14	3	ovmpt4g 7503	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → (𝑥𝐹𝑦) = 𝐶)
15	10, 11, 13, 14	syl3anc 1379	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
16	8, 15	eqtr3id 2788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
17		suppovss.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
18	17	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
19	18	mptexd 7168	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)
20		suppovss.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ 𝐶))
21	19, 20	fmptd 7055	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:𝐴⟶V)
22		ssidd 3938	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ (𝐺 supp (𝐵 × {𝑍})))
23		suppovss.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
24		snex 5368	. . . . . . . . . . . . 13 ⊢ {𝑍} ∈ V
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑍} ∈ V)
26	17, 25	xpexd 7694	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 × {𝑍}) ∈ V)
27	21, 22, 23, 26	suppssr 8135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺‘𝑥) = (𝐵 × {𝑍}))
28	27	fveq1d 6829	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺‘𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
29	12, 9, 28	syl2anc 590	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
30		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
31	20	fvmpt2 6947	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) → (𝐺‘𝑥) = (𝑦 ∈ 𝐵 ↦ 𝐶))
32	30, 19, 31	syl2anc 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝑦 ∈ 𝐵 ↦ 𝐶))
33	1	anassrs 468	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
34	32, 33	fvmpt2d 6949	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑥)‘𝑦) = 𝐶)
35	12, 10, 11, 34	syl21anc 843	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑥)‘𝑦) = 𝐶)
36		suppovss.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝐷)
37	12, 36	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑍 ∈ 𝐷)
38		fvconst2g 7146	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
39	37, 11, 38	syl2anc 590	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
40	29, 35, 39	3eqtr3d 2782	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
41	7, 16, 40	3eqtrd 2778	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = 𝑍)
42	41	adantl3r 756	. . . . 5 ⊢ (((((𝜑 ∧ 𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = 𝑍)
43		elxp2 5642	. . . . . 6 ⊢ (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
44	43	bilani 505	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
45	42, 44	r19.29vva 3199	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → (𝐹‘𝑧) = 𝑍)
46	45	adantlr 721	. . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) ∧ 𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → (𝐹‘𝑧) = 𝑍)
47		simpr 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
48	47	fveq2d 6831	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
49		simpllr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ 𝐴)
50		simplr 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))
51	50	eldifad 3895	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦 ∈ 𝐵)
52		simplll 780	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝜑)
53	52, 49, 51, 1	syl12anc 842	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 ∈ 𝐷)
54	49, 51, 53, 14	syl3anc 1379	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
55	8, 54	eqtr3id 2788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
56	52, 49, 51, 34	syl21anc 843	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑥)‘𝑦) = 𝐶)
57		fvexd 6842	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑥)‘𝑦) ∈ V)
58	33, 32, 57	fmpt2d 7066	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥):𝐵⟶V)
59		ssiun2 4977	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑥 ∈ 𝐴 ((𝐺‘𝑥) supp 𝑍))
60	59	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑥 ∈ 𝐴 ((𝐺‘𝑥) supp 𝑍))
61		fveq2 6827	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘))
62	61	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) supp 𝑍) = ((𝐺‘𝑘) supp 𝑍))
63	62	cbviunv 4968	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐺‘𝑥) supp 𝑍) = ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍)
64	60, 63	sseqtrdi 3955	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍))
65		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝜑)
66		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
67	66	eldifad 3895	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘 ∈ 𝐴)
68	21, 22, 23, 26	suppssr 8135	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺‘𝑘) = (𝐵 × {𝑍}))
69		eleq1w 2822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
70	69	anbi2d 636	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑘 ∈ 𝐴)))
71	61	fneq1d 6578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) Fn 𝐵 ↔ (𝐺‘𝑘) Fn 𝐵))
72	70, 71	imbi12d 345	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) Fn 𝐵) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐺‘𝑘) Fn 𝐵)))
73	58	ffnd 6656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) Fn 𝐵)
74	72, 73	chvarvv 1996	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐺‘𝑘) Fn 𝐵)
75	17	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑊)
76	36	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑍 ∈ 𝐷)
77		fnsuppeq0 8132	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑘) Fn 𝐵 ∧ 𝐵 ∈ 𝑊 ∧ 𝑍 ∈ 𝐷) → (((𝐺‘𝑘) supp 𝑍) = ∅ ↔ (𝐺‘𝑘) = (𝐵 × {𝑍})))
78	74, 75, 76, 77	syl3anc 1379	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (((𝐺‘𝑘) supp 𝑍) = ∅ ↔ (𝐺‘𝑘) = (𝐵 × {𝑍})))
79	78	biimpar 478	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ (𝐺‘𝑘) = (𝐵 × {𝑍})) → ((𝐺‘𝑘) supp 𝑍) = ∅)
80	65, 67, 68, 79	syl21anc 843	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺‘𝑘) supp 𝑍) = ∅)
81	80	ralrimiva 3131	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))((𝐺‘𝑘) supp 𝑍) = ∅)
82		nfcv 2901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))
83	82	iunxdif3 5024	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))((𝐺‘𝑘) supp 𝑍) = ∅ → ∪ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍))
84	81, 83	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍))
85		dfin4 4206	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
86		suppssdm 8117	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 supp (𝐵 × {𝑍})) ⊆ dom 𝐺
87	86, 21	fssdm 6674	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴)
88		sseqin2 4152	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
89	87, 88	sylib 219	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
90	85, 89	eqtr3id 2788	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) = (𝐺 supp (𝐵 × {𝑍})))
91	90	iuneq1d 4949	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
92	84, 91	eqtr3d 2776	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
93	92	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
94	64, 93	sseqtrd 3951	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
95	36	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑍 ∈ 𝐷)
96	58, 94, 18, 95	suppssr 8135	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) → ((𝐺‘𝑥)‘𝑦) = 𝑍)
97	96	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑥)‘𝑦) = 𝑍)
98	56, 97	eqtr3d 2776	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
99	48, 55, 98	3eqtrd 2778	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = 𝑍)
100	99	adantl3r 756	. . . . 5 ⊢ (((((𝜑 ∧ 𝑧 ∈ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = 𝑍)
101		elxp2 5642	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))𝑧 = ⟨𝑥, 𝑦⟩)
102	101	bilani 505	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))𝑧 = ⟨𝑥, 𝑦⟩)
103	100, 102	r19.29vva 3199	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → (𝐹‘𝑧) = 𝑍)
104	103	adantlr 721	. . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) ∧ 𝑧 ∈ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → (𝐹‘𝑧) = 𝑍)
105		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → 𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))))
106		difxp 6115	. . . . 5 ⊢ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) = (((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∪ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))))
107	105, 106	eleqtrdi 2849	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → 𝑧 ∈ (((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∪ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))))
108		elun 4083	. . . 4 ⊢ (𝑧 ∈ (((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∪ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) ↔ (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∨ 𝑧 ∈ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))))
109	107, 108	sylib 219	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∨ 𝑧 ∈ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))))
110	46, 104, 109	mpjaodan 966	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → (𝐹‘𝑧) = 𝑍)
111	5, 110	suppss 8134	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4261 {csn 4555 ⟨cop 4561 ∪ ciun 4921 ↦ cmpt 5153 × cxp 5616 Fn wfn 6480 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 supp csupp 8100

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-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 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-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 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-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-1st 7931 df-2nd 7932 df-supp 8101

This theorem is referenced by: fedgmullem1 33813

W3C validator