Theorem isssc 17872

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 17866	. . . 4 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
2		fndm 6638	. . . . . . . . . . . 12 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
3	2	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
4		isssc.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
5	4	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
6	5	fndmd 6640	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑇 × 𝑇))
7	3, 6	eqtr3d 2800	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
8	7	dmeqd 5895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom (𝑡 × 𝑡) = dom (𝑇 × 𝑇))
9		dmxpid 5920	. . . . . . . . 9 ⊢ dom (𝑡 × 𝑡) = 𝑡
10		dmxpid 5920	. . . . . . . . 9 ⊢ dom (𝑇 × 𝑇) = 𝑇
11	8, 9, 10	3eqtr3g 2821	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
12	11	ex 417	. . . . . . 7 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝑡 = 𝑇))
13		id 23	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
14	13	sqxpeqd 5693	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡 × 𝑡) = (𝑇 × 𝑇))
15	14	fneq2d 6629	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇)))
16	4, 15	syl5ibrcom 250	. . . . . . 7 ⊢ (𝜑 → (𝑡 = 𝑇 → 𝐽 Fn (𝑡 × 𝑡)))
17	12, 16	impbid 215	. . . . . 6 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝑡 = 𝑇))
18	17	anbi1d 642	. . . . 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 1951	. . . 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 286	. . 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 4576	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 𝑡 = 𝒫 𝑇)
23	22	rexeqdv 3324	. . . . 5 ⊢ (𝑡 = 𝑇 → (∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
24	23	ceqsexgv 3613	. . . 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 18	. . 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 282	. 2 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
27		df-rex 3090	. . 3 ⊢ (∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
28		3anass 1111	. . . . . . . 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 8895	. . . . . . . 8 ⊢ (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝐻 ∈ V ∧ 𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
30		vex 3459	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
31	30, 30	xpex 7748	. . . . . . . . . . 11 ⊢ (𝑠 × 𝑠) ∈ V
32		fnex 7215	. . . . . . . . . . 11 ⊢ ((𝐻 Fn (𝑠 × 𝑠) ∧ (𝑠 × 𝑠) ∈ V) → 𝐻 ∈ V)
33	31, 32	mpan2 703	. . . . . . . . . 10 ⊢ (𝐻 Fn (𝑠 × 𝑠) → 𝐻 ∈ V)
34	33	adantr 485	. . . . . . . . 9 ⊢ ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)) → 𝐻 ∈ V)
35	34	pm4.71ri 569	. . . . . . . 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 306	. . . . . . 7 ⊢ (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
37		fndm 6638	. . . . . . . . . . . . . 14 ⊢ (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
38	37	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
39		isssc.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
40	39	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
41	40	fndmd 6640	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑆 × 𝑆))
42	38, 41	eqtr3d 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
43	42	dmeqd 5895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom (𝑠 × 𝑠) = dom (𝑆 × 𝑆))
44		dmxpid 5920	. . . . . . . . . . 11 ⊢ dom (𝑠 × 𝑠) = 𝑠
45		dmxpid 5920	. . . . . . . . . . 11 ⊢ dom (𝑆 × 𝑆) = 𝑆
46	43, 44, 45	3eqtr3g 2821	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
47	46	ex 417	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 Fn (𝑠 × 𝑠) → 𝑠 = 𝑆))
48		id 23	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
49	48	sqxpeqd 5693	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠 × 𝑠) = (𝑆 × 𝑆))
50	49	fneq2d 6629	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝐻 Fn (𝑆 × 𝑆)))
51	39, 50	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 = 𝑆 → 𝐻 Fn (𝑠 × 𝑠)))
52	47, 51	impbid 215	. . . . . . . 8 ⊢ (𝜑 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝑠 = 𝑆))
53	52	anbi1d 642	. . . . . . 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 286	. . . . . 6 ⊢ (𝜑 → (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))))
55	54	anbi2d 641	. . . . 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 657	. . . . 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 290	. . . 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 1951	. . 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 286	. 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 1898	. . . . 5 ⊢ (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) → ∃𝑠 𝑠 = 𝑆)
61		isset 3469	. . . . 5 ⊢ (𝑆 ∈ V ↔ ∃𝑠 𝑠 = 𝑆)
62	60, 61	sylibr 237	. . . 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 5290	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝑆 ∈ V)
65	64	expcom 418	. . . . 5 ⊢ (𝑇 ∈ 𝑉 → (𝑆 ⊆ 𝑇 → 𝑆 ∈ V))
66	21, 65	syl 18	. . . 4 ⊢ (𝜑 → (𝑆 ⊆ 𝑇 → 𝑆 ∈ V))
67	66	adantrd 496	. . 3 ⊢ (𝜑 → ((𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → 𝑆 ∈ V))
68	30	elpw 4566	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝑇 ↔ 𝑠 ⊆ 𝑇)
69		sseq1 3962	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑇 ↔ 𝑆 ⊆ 𝑇))
70	68, 69	bitrid 286	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 𝑇 ↔ 𝑆 ⊆ 𝑇))
71	49	raleqdv 3323	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
72		fvex 6894	. . . . . . . . . 10 ⊢ (𝐻‘𝑧) ∈ V
73	72	elpw 4566	. . . . . . . . 9 ⊢ ((𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ (𝐻‘𝑧) ⊆ (𝐽‘𝑧))
74		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
75		df-ov 7413	. . . . . . . . . . 11 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
76	74, 75	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
77		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝐽‘⟨𝑥, 𝑦⟩))
78		df-ov 7413	. . . . . . . . . . 11 ⊢ (𝑥𝐽𝑦) = (𝐽‘⟨𝑥, 𝑦⟩)
79	77, 78	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝑥𝐽𝑦))
80	76, 79	sseq12d 3970	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻‘𝑧) ⊆ (𝐽‘𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
81	73, 80	bitrid 286	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
82	81	ralxp 5827	. . . . . . 7 ⊢ (∀𝑧 ∈ (𝑆 × 𝑆)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
83	71, 82	bitrdi 290	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
84	70, 83	anbi12d 643	. . . . 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 3613	. . . 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 382	. 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 308	1 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562  ⟨cop 4595   class class class wbr 5109   × cxp 5659  dom cdm 5661   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  Xcixp 8891   ⊆_cat cssc 17859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-ixp 8892  df-ssc 17862

This theorem is referenced by:  ssc1  17873  ssc2  17874  sscres  17875  ssctr  17877  0ssc  17889  catsubcat  17891  rnghmsscmap2  20728  rnghmsscmap  20729  rhmsscmap2  20757  rhmsscmap  20758  rhmsscrnghm  20764  srhmsubc  20779  fldhmsubc  20888  srhmsubcALTV  49110  fldhmsubcALTV  49118  iinfssc  49855  discsubc  49862  nelsubclem  49865  imassc  49951  setc1onsubc  50400