Theorem isssc 17532

Description: Value of the subcategory subset relation when the arguments are known functions. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
isssc.1	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
isssc.2	⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
isssc.3	⊢ (𝜑 → 𝑇 ∈ 𝑉)

Assertion

Ref	Expression
isssc	⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))

Distinct variable groups:   𝑥,𝑦,𝐻   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem isssc

Dummy variables 𝑡 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brssc 17526	. . . 4 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
2		fndm 6536	. . . . . . . . . . . 12 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
3	2	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
4		isssc.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
5	4	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
6	5	fndmd 6538	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑇 × 𝑇))
7	3, 6	eqtr3d 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
8	7	dmeqd 5814	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom (𝑡 × 𝑡) = dom (𝑇 × 𝑇))
9		dmxpid 5839	. . . . . . . . 9 ⊢ dom (𝑡 × 𝑡) = 𝑡
10		dmxpid 5839	. . . . . . . . 9 ⊢ dom (𝑇 × 𝑇) = 𝑇
11	8, 9, 10	3eqtr3g 2801	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
12	11	ex 413	. . . . . . 7 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝑡 = 𝑇))
13		id 22	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
14	13	sqxpeqd 5621	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡 × 𝑡) = (𝑇 × 𝑇))
15	14	fneq2d 6527	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇)))
16	4, 15	syl5ibrcom 246	. . . . . . 7 ⊢ (𝜑 → (𝑡 = 𝑇 → 𝐽 Fn (𝑡 × 𝑡)))
17	12, 16	impbid 211	. . . . . 6 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝑡 = 𝑇))
18	17	anbi1d 630	. . . . 5 ⊢ (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)) ↔ (𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧))))
19	18	exbidv 1924	. . . 4 ⊢ (𝜑 → (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)) ↔ ∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧))))
20	1, 19	bitrid 282	. . 3 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧))))
21		isssc.3	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
22		pweq 4549	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 𝑡 = 𝒫 𝑇)
23	22	rexeqdv 3349	. . . . 5 ⊢ (𝑡 = 𝑇 → (∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
24	23	ceqsexgv 3584	. . . 4 ⊢ (𝑇 ∈ 𝑉 → (∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
25	21, 24	syl 17	. . 3 ⊢ (𝜑 → (∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
26	20, 25	bitrd 278	. 2 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
27		df-rex 3070	. . 3 ⊢ (∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
28		3anass 1094	. . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ 𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)) ↔ (𝐻 ∈ V ∧ (𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))))
29		elixp2 8689	. . . . . . . 8 ⊢ (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝐻 ∈ V ∧ 𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
30		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
31	30, 30	xpex 7603	. . . . . . . . . . 11 ⊢ (𝑠 × 𝑠) ∈ V
32		fnex 7093	. . . . . . . . . . 11 ⊢ ((𝐻 Fn (𝑠 × 𝑠) ∧ (𝑠 × 𝑠) ∈ V) → 𝐻 ∈ V)
33	31, 32	mpan2 688	. . . . . . . . . 10 ⊢ (𝐻 Fn (𝑠 × 𝑠) → 𝐻 ∈ V)
34	33	adantr 481	. . . . . . . . 9 ⊢ ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)) → 𝐻 ∈ V)
35	34	pm4.71ri 561	. . . . . . . 8 ⊢ ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)) ↔ (𝐻 ∈ V ∧ (𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))))
36	28, 29, 35	3bitr4i 303	. . . . . . 7 ⊢ (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
37		fndm 6536	. . . . . . . . . . . . . 14 ⊢ (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
38	37	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
39		isssc.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
40	39	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
41	40	fndmd 6538	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑆 × 𝑆))
42	38, 41	eqtr3d 2780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
43	42	dmeqd 5814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom (𝑠 × 𝑠) = dom (𝑆 × 𝑆))
44		dmxpid 5839	. . . . . . . . . . 11 ⊢ dom (𝑠 × 𝑠) = 𝑠
45		dmxpid 5839	. . . . . . . . . . 11 ⊢ dom (𝑆 × 𝑆) = 𝑆
46	43, 44, 45	3eqtr3g 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
47	46	ex 413	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 Fn (𝑠 × 𝑠) → 𝑠 = 𝑆))
48		id 22	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
49	48	sqxpeqd 5621	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠 × 𝑠) = (𝑆 × 𝑆))
50	49	fneq2d 6527	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝐻 Fn (𝑆 × 𝑆)))
51	39, 50	syl5ibrcom 246	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 = 𝑆 → 𝐻 Fn (𝑠 × 𝑠)))
52	47, 51	impbid 211	. . . . . . . 8 ⊢ (𝜑 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝑠 = 𝑆))
53	52	anbi1d 630	. . . . . . 7 ⊢ (𝜑 → ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)) ↔ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))))
54	36, 53	bitrid 282	. . . . . 6 ⊢ (𝜑 → (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))))
55	54	anbi2d 629	. . . . 5 ⊢ (𝜑 → ((𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)) ↔ (𝑠 ∈ 𝒫 𝑇 ∧ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))))
56		an12 642	. . . . 5 ⊢ ((𝑠 ∈ 𝒫 𝑇 ∧ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) ↔ (𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))))
57	55, 56	bitrdi 287	. . . 4 ⊢ (𝜑 → ((𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)) ↔ (𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))))
58	57	exbidv 1924	. . 3 ⊢ (𝜑 → (∃𝑠(𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)) ↔ ∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))))
59	27, 58	bitrid 282	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ ∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))))
60		exsimpl 1871	. . . . 5 ⊢ (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) → ∃𝑠 𝑠 = 𝑆)
61		isset 3445	. . . . 5 ⊢ (𝑆 ∈ V ↔ ∃𝑠 𝑠 = 𝑆)
62	60, 61	sylibr 233	. . . 4 ⊢ (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) → 𝑆 ∈ V)
63	62	a1i 11	. . 3 ⊢ (𝜑 → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) → 𝑆 ∈ V))
64		ssexg 5247	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝑆 ∈ V)
65	64	expcom 414	. . . . 5 ⊢ (𝑇 ∈ 𝑉 → (𝑆 ⊆ 𝑇 → 𝑆 ∈ V))
66	21, 65	syl 17	. . . 4 ⊢ (𝜑 → (𝑆 ⊆ 𝑇 → 𝑆 ∈ V))
67	66	adantrd 492	. . 3 ⊢ (𝜑 → ((𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → 𝑆 ∈ V))
68	30	elpw 4537	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝑇 ↔ 𝑠 ⊆ 𝑇)
69		sseq1 3946	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑇 ↔ 𝑆 ⊆ 𝑇))
70	68, 69	bitrid 282	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 𝑇 ↔ 𝑆 ⊆ 𝑇))
71	49	raleqdv 3348	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
72		fvex 6787	. . . . . . . . . 10 ⊢ (𝐻‘𝑧) ∈ V
73	72	elpw 4537	. . . . . . . . 9 ⊢ ((𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ (𝐻‘𝑧) ⊆ (𝐽‘𝑧))
74		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
75		df-ov 7278	. . . . . . . . . . 11 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
76	74, 75	eqtr4di 2796	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
77		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝐽‘⟨𝑥, 𝑦⟩))
78		df-ov 7278	. . . . . . . . . . 11 ⊢ (𝑥𝐽𝑦) = (𝐽‘⟨𝑥, 𝑦⟩)
79	77, 78	eqtr4di 2796	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝑥𝐽𝑦))
80	76, 79	sseq12d 3954	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻‘𝑧) ⊆ (𝐽‘𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
81	73, 80	bitrid 282	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
82	81	ralxp 5750	. . . . . . 7 ⊢ (∀𝑧 ∈ (𝑆 × 𝑆)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
83	71, 82	bitrdi 287	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
84	70, 83	anbi12d 631	. . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)) ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
85	84	ceqsexgv 3584	. . . 4 ⊢ (𝑆 ∈ V → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
86	85	a1i 11	. . 3 ⊢ (𝜑 → (𝑆 ∈ V → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))))
87	63, 67, 86	pm5.21ndd 381	. 2 ⊢ (𝜑 → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
88	26, 59, 87	3bitrd 305	1 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533  ⟨cop 4567   class class class wbr 5074   × cxp 5587  dom cdm 5589   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  Xcixp 8685   ⊆_cat cssc 17519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-ixp 8686  df-ssc 17522

This theorem is referenced by:  ssc1  17533  ssc2  17534  sscres  17535  ssctr  17537  0ssc  17552  catsubcat  17554  rnghmsscmap2  45531  rnghmsscmap  45532  rhmsscmap2  45577  rhmsscmap  45578  rhmsscrnghm  45584  srhmsubc  45634  fldhmsubc  45642  srhmsubcALTV  45652  fldhmsubcALTV  45660