Theorem isssc 17781

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 17775	. . . 4 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
2		fndm 6596	. . . . . . . . . . . 12 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
3	2	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
4		isssc.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
5	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
6	5	fndmd 6598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑇 × 𝑇))
7	3, 6	eqtr3d 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
8	7	dmeqd 5855	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom (𝑡 × 𝑡) = dom (𝑇 × 𝑇))
9		dmxpid 5880	. . . . . . . . 9 ⊢ dom (𝑡 × 𝑡) = 𝑡
10		dmxpid 5880	. . . . . . . . 9 ⊢ dom (𝑇 × 𝑇) = 𝑇
11	8, 9, 10	3eqtr3g 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
12	11	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝑡 = 𝑇))
13		id 22	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
14	13	sqxpeqd 5657	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡 × 𝑡) = (𝑇 × 𝑇))
15	14	fneq2d 6587	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇)))
16	4, 15	syl5ibrcom 247	. . . . . . 7 ⊢ (𝜑 → (𝑡 = 𝑇 → 𝐽 Fn (𝑡 × 𝑡)))
17	12, 16	impbid 212	. . . . . 6 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝑡 = 𝑇))
18	17	anbi1d 632	. . . . 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 1923	. . . 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 283	. . 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 4556	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 𝑡 = 𝒫 𝑇)
23	22	rexeqdv 3297	. . . . 5 ⊢ (𝑡 = 𝑇 → (∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
24	23	ceqsexgv 3597	. . . 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 279	. 2 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
27		df-rex 3063	. . 3 ⊢ (∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
28		3anass 1095	. . . . . . . 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 8843	. . . . . . . 8 ⊢ (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝐻 ∈ V ∧ 𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
30		vex 3434	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
31	30, 30	xpex 7701	. . . . . . . . . . 11 ⊢ (𝑠 × 𝑠) ∈ V
32		fnex 7166	. . . . . . . . . . 11 ⊢ ((𝐻 Fn (𝑠 × 𝑠) ∧ (𝑠 × 𝑠) ∈ V) → 𝐻 ∈ V)
33	31, 32	mpan2 692	. . . . . . . . . 10 ⊢ (𝐻 Fn (𝑠 × 𝑠) → 𝐻 ∈ V)
34	33	adantr 480	. . . . . . . . 9 ⊢ ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)) → 𝐻 ∈ V)
35	34	pm4.71ri 560	. . . . . . . 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 6596	. . . . . . . . . . . . . 14 ⊢ (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
38	37	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
39		isssc.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
40	39	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
41	40	fndmd 6598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑆 × 𝑆))
42	38, 41	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
43	42	dmeqd 5855	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom (𝑠 × 𝑠) = dom (𝑆 × 𝑆))
44		dmxpid 5880	. . . . . . . . . . 11 ⊢ dom (𝑠 × 𝑠) = 𝑠
45		dmxpid 5880	. . . . . . . . . . 11 ⊢ dom (𝑆 × 𝑆) = 𝑆
46	43, 44, 45	3eqtr3g 2795	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
47	46	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 Fn (𝑠 × 𝑠) → 𝑠 = 𝑆))
48		id 22	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
49	48	sqxpeqd 5657	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠 × 𝑠) = (𝑆 × 𝑆))
50	49	fneq2d 6587	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝐻 Fn (𝑆 × 𝑆)))
51	39, 50	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 = 𝑆 → 𝐻 Fn (𝑠 × 𝑠)))
52	47, 51	impbid 212	. . . . . . . 8 ⊢ (𝜑 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝑠 = 𝑆))
53	52	anbi1d 632	. . . . . . 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 283	. . . . . 6 ⊢ (𝜑 → (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))))
55	54	anbi2d 631	. . . . 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 646	. . . . 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 1923	. . 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 283	. 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 1870	. . . . 5 ⊢ (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧))) → ∃𝑠 𝑠 = 𝑆)
61		isset 3444	. . . . 5 ⊢ (𝑆 ∈ V ↔ ∃𝑠 𝑠 = 𝑆)
62	60, 61	sylibr 234	. . . 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 5261	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝑆 ∈ V)
65	64	expcom 413	. . . . 5 ⊢ (𝑇 ∈ 𝑉 → (𝑆 ⊆ 𝑇 → 𝑆 ∈ V))
66	21, 65	syl 17	. . . 4 ⊢ (𝜑 → (𝑆 ⊆ 𝑇 → 𝑆 ∈ V))
67	66	adantrd 491	. . 3 ⊢ (𝜑 → ((𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → 𝑆 ∈ V))
68	30	elpw 4546	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝑇 ↔ 𝑠 ⊆ 𝑇)
69		sseq1 3948	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑇 ↔ 𝑆 ⊆ 𝑇))
70	68, 69	bitrid 283	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 𝑇 ↔ 𝑆 ⊆ 𝑇))
71	49	raleqdv 3296	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
72		fvex 6848	. . . . . . . . . 10 ⊢ (𝐻‘𝑧) ∈ V
73	72	elpw 4546	. . . . . . . . 9 ⊢ ((𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ (𝐻‘𝑧) ⊆ (𝐽‘𝑧))
74		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
75		df-ov 7364	. . . . . . . . . . 11 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
76	74, 75	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
77		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝐽‘⟨𝑥, 𝑦⟩))
78		df-ov 7364	. . . . . . . . . . 11 ⊢ (𝑥𝐽𝑦) = (𝐽‘⟨𝑥, 𝑦⟩)
79	77, 78	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝑥𝐽𝑦))
80	76, 79	sseq12d 3956	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻‘𝑧) ⊆ (𝐽‘𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
81	73, 80	bitrid 283	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
82	81	ralxp 5791	. . . . . . 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 633	. . . . 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 3597	. . . 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 379	. 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  𝒫 cpw 4542  ⟨cop 4574   class class class wbr 5086   × cxp 5623  dom cdm 5625   Fn wfn 6488  ‘cfv 6493  (class class class)co 7361  Xcixp 8839   ⊆_cat cssc 17768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-ixp 8840  df-ssc 17771

This theorem is referenced by:  ssc1  17782  ssc2  17783  sscres  17784  ssctr  17786  0ssc  17798  catsubcat  17800  rnghmsscmap2  20600  rnghmsscmap  20601  rhmsscmap2  20629  rhmsscmap  20630  rhmsscrnghm  20636  srhmsubc  20651  fldhmsubc  20756  srhmsubcALTV  48816  fldhmsubcALTV  48824  iinfssc  49547  discsubc  49554  nelsubclem  49557  imassc  49643  setc1onsubc  50092