Theorem xkocnv 21617

Description: The inverse of the "currying" function 𝐹 is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)

Hypotheses

Ref	Expression
xkohmeo.x	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
xkohmeo.y	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
xkohmeo.f	⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))
xkohmeo.j	⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp)
xkohmeo.k	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
xkohmeo.l	⊢ (𝜑 → 𝐿 ∈ Top)

Assertion

Ref	Expression
xkocnv	⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝐽   𝑓,𝐾,𝑔,𝑥,𝑦   𝜑,𝑓,𝑔,𝑥,𝑦   𝑓,𝐿,𝑔,𝑥,𝑦   𝑓,𝑋,𝑔,𝑥,𝑦   𝑓,𝑌,𝑔,𝑥,𝑦   𝑓,𝐹,𝑔,𝑥,𝑦

Proof of Theorem xkocnv

Step	Hyp	Ref	Expression
1		simprr 796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))
2		xkohmeo.x	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝐽 ∈ (TopOn‘𝑋))
4		xkohmeo.y	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
5	4	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑌))
6		txtopon 21394	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7	2, 4, 6	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
8	7	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
9		xkohmeo.l	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ∈ Top)
10		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐿 = ∪ 𝐿
11	10	toptopon 20722	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
12	9, 11	sylib 208	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
13	12	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝐿 ∈ (TopOn‘∪ 𝐿))
14		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
15		cnf2 21053	. . . . . . . . . . . 12 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓:(𝑋 × 𝑌)⟶∪ 𝐿)
16	8, 13, 14, 15	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓:(𝑋 × 𝑌)⟶∪ 𝐿)
17		ffn 6045	. . . . . . . . . . 11 ⊢ (𝑓:(𝑋 × 𝑌)⟶∪ 𝐿 → 𝑓 Fn (𝑋 × 𝑌))
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓 Fn (𝑋 × 𝑌))
19		fnov 6768	. . . . . . . . . 10 ⊢ (𝑓 Fn (𝑋 × 𝑌) ↔ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
20	18, 19	sylib 208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
21	20, 14	eqeltrrd 2702	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
22	3, 5, 21	cnmpt2k 21491	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
23	22	adantrr 753	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
24	1, 23	eqeltrd 2701	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
25	20	adantrr 753	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
26		eqid 2622	. . . . . . 7 ⊢ 𝑋 = 𝑋
27		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜑
28		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)
29		nfmpt1 4747	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
30	29	nfeq2 2780	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
31	28, 30	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))
32	27, 31	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))))
33		nfv 1843	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝜑
34		nfv 1843	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)
35		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦𝑋
36		nfmpt1 4747	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))
37	35, 36	nfmpt 4746	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
38	37	nfeq2 2780	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
39	34, 38	nfan 1828	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))
40	33, 39	nfan 1828	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))))
41		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑥 ∈ 𝑋
42	40, 41	nfan 1828	. . . . . . . . . . 11 ⊢ Ⅎ𝑦((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ 𝑥 ∈ 𝑋)
43		simplrr 801	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))
44	43	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑔‘𝑥) = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))‘𝑥))
45		simprl 794	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑋)
46		toponmax 20730	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
47	4, 46	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
48	47	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑌 ∈ 𝐾)
49		mptexg 6484	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ 𝐾 → (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) ∈ V)
50	48, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) ∈ V)
51		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
52	51	fvmpt2 6291	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) ∈ V) → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))‘𝑥) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
53	45, 50, 52	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))‘𝑥) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
54	44, 53	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑔‘𝑥) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
55	54	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑔‘𝑥)‘𝑦) = ((𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))‘𝑦))
56		simprr 796	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
57		ovex 6678	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑓𝑦) ∈ V
58		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))
59	58	fvmpt2 6291	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑌 ∧ (𝑥𝑓𝑦) ∈ V) → ((𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))‘𝑦) = (𝑥𝑓𝑦))
60	56, 57, 59	sylancl 694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))‘𝑦) = (𝑥𝑓𝑦))
61	55, 60	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦))
62	61	expr 643	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 → ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦)))
63	42, 62	ralrimi 2957	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦))
64		eqid 2622	. . . . . . . . . 10 ⊢ 𝑌 = 𝑌
65	63, 64	jctil 560	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ 𝑥 ∈ 𝑋) → (𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦)))
66	65	ex 450	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → (𝑥 ∈ 𝑋 → (𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦))))
67	32, 66	ralrimi 2957	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → ∀𝑥 ∈ 𝑋 (𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦)))
68		mpt2eq123 6714	. . . . . . 7 ⊢ ((𝑋 = 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
69	26, 67, 68	sylancr 695	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
70	25, 69	eqtr4d 2659	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))
71	24, 70	jca 554	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
72		simprr 796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))
73	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝐽 ∈ (TopOn‘𝑋))
74	4	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝐾 ∈ (TopOn‘𝑌))
75	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝐿 ∈ (TopOn‘∪ 𝐿))
76		xkohmeo.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
77	76	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝐾 ∈ 𝑛-Locally Comp)
78		nllytop 21276	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
79	77, 78	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝐾 ∈ Top)
80	9	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝐿 ∈ Top)
81		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
82	81	xkotopon 21403	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
83	79, 80, 82	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
84		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
85		cnf2 21053	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝑔:𝑋⟶(𝐾 Cn 𝐿))
86	73, 83, 84, 85	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝑔:𝑋⟶(𝐾 Cn 𝐿))
87	86	feqmptd 6249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑔‘𝑥)))
88	4	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
89	12	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘∪ 𝐿))
90	86	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → (𝑔‘𝑥) ∈ (𝐾 Cn 𝐿))
91		cnf2 21053	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑔‘𝑥) ∈ (𝐾 Cn 𝐿)) → (𝑔‘𝑥):𝑌⟶∪ 𝐿)
92	88, 89, 90, 91	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → (𝑔‘𝑥):𝑌⟶∪ 𝐿)
93	92	feqmptd 6249	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → (𝑔‘𝑥) = (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))
94	93	mpteq2dva 4744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (𝑥 ∈ 𝑋 ↦ (𝑔‘𝑥)) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
95	87, 94	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
96	95, 84	eqeltrrd 2702	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
97	73, 74, 75, 77, 96	cnmptk2 21489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
98	97	adantrr 753	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
99	72, 98	eqeltrd 2701	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
100	95	adantrr 753	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
101		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))
102		nfmpt21 6722	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))
103	102	nfeq2 2780	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))
104	101, 103	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))
105	27, 104	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
106		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾))
107		nfmpt22 6723	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))
108	107	nfeq2 2780	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))
109	106, 108	nfan 1828	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))
110	33, 109	nfan 1828	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
111	110, 41	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ 𝑥 ∈ 𝑋)
112	72	oveqd 6667	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → (𝑥𝑓𝑦) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))𝑦))
113		fvex 6201	. . . . . . . . . . . 12 ⊢ ((𝑔‘𝑥)‘𝑦) ∈ V
114		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))
115	114	ovmpt4g 6783	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ ((𝑔‘𝑥)‘𝑦) ∈ V) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))𝑦) = ((𝑔‘𝑥)‘𝑦))
116	113, 115	mp3an3 1413	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))𝑦) = ((𝑔‘𝑥)‘𝑦))
117	112, 116	sylan9eq 2676	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑓𝑦) = ((𝑔‘𝑥)‘𝑦))
118	117	expr 643	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 → (𝑥𝑓𝑦) = ((𝑔‘𝑥)‘𝑦)))
119	111, 118	ralrimi 2957	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 (𝑥𝑓𝑦) = ((𝑔‘𝑥)‘𝑦))
120		mpteq12 4736	. . . . . . . 8 ⊢ ((𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 (𝑥𝑓𝑦) = ((𝑔‘𝑥)‘𝑦)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) = (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))
121	64, 119, 120	sylancr 695	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) = (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))
122	105, 121	mpteq2da 4743	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
123	100, 122	eqtr4d 2659	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))
124	99, 123	jca 554	. . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))))
125	71, 124	impbida 877	. . 3 ⊢ (𝜑 → ((𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) ↔ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))))
126	125	opabbidv 4716	. 2 ⊢ (𝜑 → {⟨𝑔, 𝑓⟩ ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))} = {⟨𝑔, 𝑓⟩ ∣ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))})
127		xkohmeo.f	. . . . 5 ⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))
128		df-mpt 4730	. . . . 5 ⊢ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) = {⟨𝑓, 𝑔⟩ ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))}
129	127, 128	eqtri 2644	. . . 4 ⊢ 𝐹 = {⟨𝑓, 𝑔⟩ ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))}
130	129	cnveqi 5297	. . 3 ⊢ ◡𝐹 = ◡{⟨𝑓, 𝑔⟩ ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))}
131		cnvopab 5533	. . 3 ⊢ ◡{⟨𝑓, 𝑔⟩ ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))} = {⟨𝑔, 𝑓⟩ ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))}
132	130, 131	eqtri 2644	. 2 ⊢ ◡𝐹 = {⟨𝑔, 𝑓⟩ ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))}
133		df-mpt 4730	. 2 ⊢ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) = {⟨𝑔, 𝑓⟩ ∣ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))}
134	126, 132, 133	3eqtr4g 2681	1 ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  ∪ cuni 4436  {copab 4712   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Topctop 20698  TopOnctopon 20715   Cn ccn 21028  Compccmp 21189  𝑛-Locally cnlly 21268   ×_t ctx 21363   ^_ko cxko 21364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-pt 16105  df-top 20699  df-topon 20716  df-bases 20750  df-ntr 20824  df-nei 20902  df-cn 21031  df-cnp 21032  df-cmp 21190  df-nlly 21270  df-tx 21365  df-xko 21366

This theorem is referenced by:  xkohmeo  21618