suppovss - Mathbox for Thierry Arnoux

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Theorem suppovss 30691

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 3102	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
3		suppovss.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	fmpo 7816	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷)
5	2, 4	sylib 221	. 2 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)⟶𝐷)
6		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
7	6	fveq2d 6699	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
8		df-ov 7194	. . . . . . . 8 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
9		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
10	9	eldifad 3865	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ 𝐴)
11		simplr 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦 ∈ 𝐵)
12		simplll 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝜑)
13	12, 10, 11, 1	syl12anc 837	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 ∈ 𝐷)
14	3	ovmpt4g 7334	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → (𝑥𝐹𝑦) = 𝐶)
15	10, 11, 13, 14	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
16	8, 15	eqtr3id 2785	. . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
19	18	mptexd 7018	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)
20		suppovss.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ 𝐶))
21	19, 20	fmptd 6909	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:𝐴⟶V)
22		ssidd 3910	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ (𝐺 supp (𝐵 × {𝑍})))
23		suppovss.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
24		snex 5309	. . . . . . . . . . . . 13 ⊢ {𝑍} ∈ V
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑍} ∈ V)
26	17, 25	xpexd 7514	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 × {𝑍}) ∈ V)
27	21, 22, 23, 26	suppssr 7916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺‘𝑥) = (𝐵 × {𝑍}))
28	27	fveq1d 6697	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺‘𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
29	12, 9, 28	syl2anc 587	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
30		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
31	20	fvmpt2 6807	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) → (𝐺‘𝑥) = (𝑦 ∈ 𝐵 ↦ 𝐶))
32	30, 19, 31	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝑦 ∈ 𝐵 ↦ 𝐶))
33	1	anassrs 471	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
34	32, 33	fvmpt2d 6809	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑥)‘𝑦) = 𝐶)
35	12, 10, 11, 34	syl21anc 838	. . . . . . . 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 6995	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
39	37, 11, 38	syl2anc 587	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
40	29, 35, 39	3eqtr3d 2779	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
41	7, 16, 40	3eqtrd 2775	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = 𝑍)
42	41	adantl3r 750	. . . . 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 5560	. . . . . . 7 ⊢ (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
44	43	biimpi 219	. . . . . 6 ⊢ (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) → ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
45	44	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
46	42, 45	r19.29vva 3242	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → (𝐹‘𝑧) = 𝑍)
47	46	adantlr 715	. . 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)
48		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
49	48	fveq2d 6699	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
50		simpllr 776	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ 𝐴)
51		simplr 769	. . . . . . . . . 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)))
52	51	eldifad 3865	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦 ∈ 𝐵)
53		simplll 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝜑)
54	53, 50, 52, 1	syl12anc 837	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 ∈ 𝐷)
55	50, 52, 54, 14	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
56	8, 55	eqtr3id 2785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
57	53, 50, 52, 34	syl21anc 838	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑥)‘𝑦) = 𝐶)
58		fvexd 6710	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑥)‘𝑦) ∈ V)
59	33, 32, 58	fmpt2d 6918	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥):𝐵⟶V)
60		ssiun2 4942	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑥 ∈ 𝐴 ((𝐺‘𝑥) supp 𝑍))
61	60	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑥 ∈ 𝐴 ((𝐺‘𝑥) supp 𝑍))
62		fveq2 6695	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘))
63	62	oveq1d 7206	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) supp 𝑍) = ((𝐺‘𝑘) supp 𝑍))
64	63	cbviunv 4935	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐺‘𝑥) supp 𝑍) = ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍)
65	61, 64	sseqtrdi 3937	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍))
66		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝜑)
67		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
68	67	eldifad 3865	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘 ∈ 𝐴)
69	21, 22, 23, 26	suppssr 7916	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺‘𝑘) = (𝐵 × {𝑍}))
70		eleq1w 2813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
71	70	anbi2d 632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑘 ∈ 𝐴)))
72	62	fneq1d 6450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) Fn 𝐵 ↔ (𝐺‘𝑘) Fn 𝐵))
73	71, 72	imbi12d 348	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) Fn 𝐵) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐺‘𝑘) Fn 𝐵)))
74	59	ffnd 6524	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) Fn 𝐵)
75	73, 74	chvarvv 2008	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐺‘𝑘) Fn 𝐵)
76	17	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑊)
77	36	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑍 ∈ 𝐷)
78		fnsuppeq0 7912	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑘) Fn 𝐵 ∧ 𝐵 ∈ 𝑊 ∧ 𝑍 ∈ 𝐷) → (((𝐺‘𝑘) supp 𝑍) = ∅ ↔ (𝐺‘𝑘) = (𝐵 × {𝑍})))
79	75, 76, 77, 78	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (((𝐺‘𝑘) supp 𝑍) = ∅ ↔ (𝐺‘𝑘) = (𝐵 × {𝑍})))
80	79	biimpar 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ (𝐺‘𝑘) = (𝐵 × {𝑍})) → ((𝐺‘𝑘) supp 𝑍) = ∅)
81	66, 68, 69, 80	syl21anc 838	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺‘𝑘) supp 𝑍) = ∅)
82	81	ralrimiva 3095	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))((𝐺‘𝑘) supp 𝑍) = ∅)
83		nfcv 2897	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))
84	83	iunxdif3 4989	. . . . . . . . . . . . . 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))
85	82, 84	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍))
86		dfin4 4168	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
87		suppssdm 7897	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 supp (𝐵 × {𝑍})) ⊆ dom 𝐺
88	87, 21	fssdm 6543	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴)
89		sseqin2 4116	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
90	88, 89	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
91	86, 90	eqtr3id 2785	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) = (𝐺 supp (𝐵 × {𝑍})))
92	91	iuneq1d 4917	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
93	85, 92	eqtr3d 2773	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
94	93	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ 𝑘 ∈ 𝐴 ((𝐺‘𝑘) supp 𝑍) = ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
95	65, 94	sseqtrd 3927	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) supp 𝑍) ⊆ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))
96	36	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑍 ∈ 𝐷)
97	59, 95, 18, 96	suppssr 7916	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) → ((𝐺‘𝑥)‘𝑦) = 𝑍)
98	97	adantr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑥)‘𝑦) = 𝑍)
99	57, 98	eqtr3d 2773	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
100	49, 56, 99	3eqtrd 2775	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑧) = 𝑍)
101	100	adantl3r 750	. . . . 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)
102		elxp2 5560	. . . . . . 7 ⊢ (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	102	biimpi 219	. . . . . 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⟩)
104	103	adantl 485	. . . . 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⟩)
105	101, 104	r19.29vva 3242	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × (𝐵 ∖ ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → (𝐹‘𝑧) = 𝑍)
106	105	adantlr 715	. . 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)
107		simpr 488	. . . . 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))))
108		difxp 6007	. . . . 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))))
109	107, 108	eleqtrdi 2841	. . . 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)))))
110		elun 4049	. . . 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)))))
111	109, 110	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)))))
112	47, 106, 111	mpjaodan 959	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))) → (𝐹‘𝑧) = 𝑍)
113	5, 112	suppss 7914	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 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 Vcvv 3398 ∖ cdif 3850 ∪ cun 3851 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 {csn 4527 ⟨cop 4533 ∪ ciun 4890 ↦ cmpt 5120 × cxp 5534 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 supp csupp 7881

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-supp 7882

This theorem is referenced by: fedgmullem1 31378

W3C validator