Theorem isssc 17901

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 17895	. . . 4 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
2		fndm 6642	. . . . . . . . . . . 12 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
3	2	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
4		isssc.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
5	4	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
6	5	fndmd 6644	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑇 × 𝑇))
7	3, 6	eqtr3d 2802	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
8	7	dmeqd 5897	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom (𝑡 × 𝑡) = dom (𝑇 × 𝑇))
9		dmxpid 5922	. . . . . . . . 9 ⊢ dom (𝑡 × 𝑡) = 𝑡
10		dmxpid 5922	. . . . . . . . 9 ⊢ dom (𝑇 × 𝑇) = 𝑇
11	8, 9, 10	3eqtr3g 2823	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
12	11	ex 418	. . . . . . 7 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝑡 = 𝑇))
13		id 23	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
14	13	sqxpeqd 5695	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡 × 𝑡) = (𝑇 × 𝑇))
15	14	fneq2d 6633	. . . . . . . 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 643	. . . . 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 1954	. . . 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 4578	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 𝑡 = 𝒫 𝑇)
23	22	rexeqdv 3326	. . . . 5 ⊢ (𝑡 = 𝑇 → (∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
24	23	ceqsexgv 3615	. . . 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 3092	. . 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 8905	. . . . . . . 8 ⊢ (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝐻 ∈ V ∧ 𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
30		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
31	30, 30	xpex 7758	. . . . . . . . . . 11 ⊢ (𝑠 × 𝑠) ∈ V
32		fnex 7222	. . . . . . . . . . 11 ⊢ ((𝐻 Fn (𝑠 × 𝑠) ∧ (𝑠 × 𝑠) ∈ V) → 𝐻 ∈ V)
33	31, 32	mpan2 704	. . . . . . . . . 10 ⊢ (𝐻 Fn (𝑠 × 𝑠) → 𝐻 ∈ V)
34	33	adantr 486	. . . . . . . . 9 ⊢ ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)) → 𝐻 ∈ V)
35	34	pm4.71ri 570	. . . . . . . 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 6642	. . . . . . . . . . . . . 14 ⊢ (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
38	37	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
39		isssc.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
40	39	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
41	40	fndmd 6644	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑆 × 𝑆))
42	38, 41	eqtr3d 2802	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
43	42	dmeqd 5897	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom (𝑠 × 𝑠) = dom (𝑆 × 𝑆))
44		dmxpid 5922	. . . . . . . . . . 11 ⊢ dom (𝑠 × 𝑠) = 𝑠
45		dmxpid 5922	. . . . . . . . . . 11 ⊢ dom (𝑆 × 𝑆) = 𝑆
46	43, 44, 45	3eqtr3g 2823	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
47	46	ex 418	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 Fn (𝑠 × 𝑠) → 𝑠 = 𝑆))
48		id 23	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
49	48	sqxpeqd 5695	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠 × 𝑠) = (𝑆 × 𝑆))
50	49	fneq2d 6633	. . . . . . . . . 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 643	. . . . . . 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 642	. . . . 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 658	. . . . 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 1954	. . 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 1901	. . . . 5 ⊢ (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) → ∃𝑠 𝑠 = 𝑆)
61		isset 3471	. . . . 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 5292	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝑆 ∈ V)
65	64	expcom 419	. . . . 5 ⊢ (𝑇 ∈ 𝑉 → (𝑆 ⊆ 𝑇 → 𝑆 ∈ V))
66	21, 65	syl 18	. . . 4 ⊢ (𝜑 → (𝑆 ⊆ 𝑇 → 𝑆 ∈ V))
67	66	adantrd 497	. . 3 ⊢ (𝜑 → ((𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → 𝑆 ∈ V))
68	30	elpw 4568	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝑇 ↔ 𝑠 ⊆ 𝑇)
69		sseq1 3963	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑇 ↔ 𝑆 ⊆ 𝑇))
70	68, 69	bitrid 286	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 𝑇 ↔ 𝑆 ⊆ 𝑇))
71	49	raleqdv 3325	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
72		fvex 6898	. . . . . . . . . 10 ⊢ (𝐻‘𝑧) ∈ V
73	72	elpw 4568	. . . . . . . . 9 ⊢ ((𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ (𝐻‘𝑧) ⊆ (𝐽‘𝑧))
74		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
75		df-ov 7422	. . . . . . . . . . 11 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
76	74, 75	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
77		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝐽‘⟨𝑥, 𝑦⟩))
78		df-ov 7422	. . . . . . . . . . 11 ⊢ (𝑥𝐽𝑦) = (𝐽‘⟨𝑥, 𝑦⟩)
79	77, 78	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝑥𝐽𝑦))
80	76, 79	sseq12d 3971	. . . . . . . . 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 5829	. . . . . . 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 644	. . . . 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 3615	. . . 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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  𝒫 cpw 4564  ⟨cop 4597   class class class wbr 5111   × cxp 5661  dom cdm 5663   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419  Xcixp 8901   ⊆_cat cssc 17888

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-ixp 8902  df-ssc 17891

This theorem is used by:  ssc1  17902  ssc2  17903  sscres  17904  ssctr  17906  0ssc  17918  catsubcat  17920  rnghmsscmap2  20780  rnghmsscmap  20781  rhmsscmap2  20809  rhmsscmap  20810  rhmsscrnghm  20816  srhmsubc  20831  fldhmsubc  20940  srhmsubcALTV  49149  fldhmsubcALTV  49157  iinfssc  49894  discsubc  49901  nelsubclem  49904  imassc  49990  setc1onsubc  50439