suppovss - Mathbox for Thierry Arnoux

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Theorem suppovss 33099

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 3210	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
3		suppovss.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	fmpo 8071	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷)
5	2, 4	sylib 221	. 2 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)⟶𝐷)
6		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
7	6	fveq2d 6889	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
8		df-ov 7422	. . . . . . . 8 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
9		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
10	9	eldifad 3918	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ 𝐴)
11		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦 ∈ 𝐵)
12		simplll 787	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝜑)
13	12, 10, 11, 1	syl12anc 850	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 ∈ 𝐷)
14	3	ovmpt4g 7566	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → (𝑥𝐹𝑦) = 𝐶)
15	10, 11, 13, 14	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
16	8, 15	eqtr3id 2814	. . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
19	18	mptexd 7229	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)
20		suppovss.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ 𝐶))
21	19, 20	fmptd 7113	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:𝐴⟶V)
22		ssidd 3961	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ (𝐺 supp (𝐵 × {𝑍})))
23		suppovss.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
24		snex 5412	. . . . . . . . . . . . 13 ⊢ {𝑍} ∈ V
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑍} ∈ V)
26	17, 25	xpexd 7756	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 × {𝑍}) ∈ V)
27	21, 22, 23, 26	suppssr 8197	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺‘𝑥) = (𝐵 × {𝑍}))
28	27	fveq1d 6887	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺‘𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
29	12, 9, 28	syl2anc 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
30		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
31	20	fvmpt2 7005	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) → (𝐺‘𝑥) = (𝑦 ∈ 𝐵 ↦ 𝐶))
32	30, 19, 31	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝑦 ∈ 𝐵 ↦ 𝐶))
33	1	anassrs 473	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
34	32, 33	fvmpt2d 7007	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑥)‘𝑦) = 𝐶)
35	12, 10, 11, 34	syl21anc 851	. . . . . . . 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 18	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑍 ∈ 𝐷)
38		fvconst2g 7207	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
39	37, 11, 38	syl2anc 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
40	29, 35, 39	3eqtr3d 2808	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
41	7, 16, 40	3eqtrd 2804	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = 𝑍)
42	41	adantl3r 763	. . . . 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 5687	. . . . . 6 ⊢ (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
44	43	bilani 510	. . . . 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 3227	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → (𝐹‘𝑧) = 𝑍)
46	45	adantlr 728	. . 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 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
48	47	fveq2d 6889	. . . . . . 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 788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ 𝐴)
50		simplr 781	. . . . . . . . . 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 3918	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦 ∈ 𝐵)
52		simplll 787	. . . . . . . . . 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 850	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
55	8, 54	eqtr3id 2814	. . . . . . 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 851	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑥)‘𝑦) = 𝐶)
57		fvexd 6900	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑥)‘𝑦) ∈ V)
58	33, 32, 57	fmpt2d 7124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥):𝐵⟶V)
59		ssiun2 5014	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑥 ∈ 𝐴 ((𝐺‘𝑥) supp 𝑍))
60	59	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑥 ∈ 𝐴 ((𝐺‘𝑥) supp 𝑍))
61		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘))
62	61	oveq1d 7434	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) supp 𝑍) = ((𝐺‘𝑘) supp 𝑍))
63	62	cbviunv 5005	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐺‘𝑥) supp 𝑍) = ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍)
64	60, 63	sseqtrdi 3978	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍))
65		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝜑)
66		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
67	66	eldifad 3918	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘 ∈ 𝐴)
68	21, 22, 23, 26	suppssr 8197	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺‘𝑘) = (𝐵 × {𝑍}))
69		eleq1w 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
70	69	anbi2d 642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑘 ∈ 𝐴)))
71	61	fneq1d 6632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) Fn 𝐵 ↔ (𝐺‘𝑘) Fn 𝐵))
72	70, 71	imbi12d 347	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) Fn 𝐵) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐺‘𝑘) Fn 𝐵)))
73	58	ffnd 6710	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) Fn 𝐵)
74	72, 73	chvarvv 2022	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐺‘𝑘) Fn 𝐵)
75	17	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑊)
76	36	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑍 ∈ 𝐷)
77		fnsuppeq0 8194	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑘) Fn 𝐵 ∧ 𝐵 ∈ 𝑊 ∧ 𝑍 ∈ 𝐷) → (((𝐺‘𝑘) supp 𝑍) = ∅ ↔ (𝐺‘𝑘) = (𝐵 × {𝑍})))
78	74, 75, 76, 77	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (((𝐺‘𝑘) supp 𝑍) = ∅ ↔ (𝐺‘𝑘) = (𝐵 × {𝑍})))
79	78	biimpar 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ (𝐺‘𝑘) = (𝐵 × {𝑍})) → ((𝐺‘𝑘) supp 𝑍) = ∅)
80	65, 67, 68, 79	syl21anc 851	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺‘𝑘) supp 𝑍) = ∅)
81	80	ralrimiva 3159	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))((𝐺‘𝑘) supp 𝑍) = ∅)
82		nfcv 2927	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))
83	82	iunxdif3 5063	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍))
85		dfin4 4231	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
86		suppssdm 8179	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 supp (𝐵 × {𝑍})) ⊆ dom 𝐺
87	86, 21	fssdm 6729	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴)
88		sseqin2 4176	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
89	87, 88	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
90	85, 89	eqtr3id 2814	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) = (𝐺 supp (𝐵 × {𝑍})))
91	90	iuneq1d 4986	. . . . . . . . . . . . 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 2802	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
93	92	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
94	64, 93	sseqtrd 3974	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
95	36	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑍 ∈ 𝐷)
96	58, 94, 18, 95	suppssr 8197	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) → ((𝐺‘𝑥)‘𝑦) = 𝑍)
97	96	adantr 486	. . . . . . . 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 2802	. . . . . . 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 2804	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = 𝑍)
100	99	adantl3r 763	. . . . 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 5687	. . . . . 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 510	. . . . 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 3227	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → (𝐹‘𝑧) = 𝑍)
104	103	adantlr 728	. . 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 490	. . . . 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 6163	. . . . 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 2875	. . . 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 4107	. . . 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 221	. . 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 973	. 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 8196	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 Vcvv 3457 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 {csn 4591 ⟨cop 4597 ∪ ciun 4958 ↦ cmpt 5194 × cxp 5661 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 supp csupp 8162

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7992 df-2nd 7993 df-supp 8163

This theorem is used by: fedgmullem1 34085

W3C validator