Theorem isssc 17787

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 17781	. . . 4 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
2		fndm 6601	. . . . . . . . . . . 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 6603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑇 × 𝑇))
7	3, 6	eqtr3d 2773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
8	7	dmeqd 5860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom (𝑡 × 𝑡) = dom (𝑇 × 𝑇))
9		dmxpid 5885	. . . . . . . . 9 ⊢ dom (𝑡 × 𝑡) = 𝑡
10		dmxpid 5885	. . . . . . . . 9 ⊢ dom (𝑇 × 𝑇) = 𝑇
11	8, 9, 10	3eqtr3g 2794	. . . . . . . 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 5663	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡 × 𝑡) = (𝑇 × 𝑇))
15	14	fneq2d 6592	. . . . . . . 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 4555	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 𝑡 = 𝒫 𝑇)
23	22	rexeqdv 3296	. . . . 5 ⊢ (𝑡 = 𝑇 → (∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧)))
24	23	ceqsexgv 3596	. . . 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 3062	. . 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 8849	. . . . . . . 8 ⊢ (𝐻 ∈ X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑧) ↔ (𝐻 ∈ V ∧ 𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
30		vex 3433	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
31	30, 30	xpex 7707	. . . . . . . . . . 11 ⊢ (𝑠 × 𝑠) ∈ V
32		fnex 7172	. . . . . . . . . . 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 6601	. . . . . . . . . . . . . 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 6603	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑆 × 𝑆))
42	38, 41	eqtr3d 2773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
43	42	dmeqd 5860	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom (𝑠 × 𝑠) = dom (𝑆 × 𝑆))
44		dmxpid 5885	. . . . . . . . . . 11 ⊢ dom (𝑠 × 𝑠) = 𝑠
45		dmxpid 5885	. . . . . . . . . . 11 ⊢ dom (𝑆 × 𝑆) = 𝑆
46	43, 44, 45	3eqtr3g 2794	. . . . . . . . . 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 5663	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠 × 𝑠) = (𝑆 × 𝑆))
50	49	fneq2d 6592	. . . . . . . . . 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 3443	. . . . 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 5264	. . . . . 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 4545	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝑇 ↔ 𝑠 ⊆ 𝑇)
69		sseq1 3947	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑇 ↔ 𝑆 ⊆ 𝑇))
70	68, 69	bitrid 283	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 𝑇 ↔ 𝑆 ⊆ 𝑇))
71	49	raleqdv 3295	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)(𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧)))
72		fvex 6853	. . . . . . . . . 10 ⊢ (𝐻‘𝑧) ∈ V
73	72	elpw 4545	. . . . . . . . 9 ⊢ ((𝐻‘𝑧) ∈ 𝒫 (𝐽‘𝑧) ↔ (𝐻‘𝑧) ⊆ (𝐽‘𝑧))
74		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
75		df-ov 7370	. . . . . . . . . . 11 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
76	74, 75	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
77		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝐽‘⟨𝑥, 𝑦⟩))
78		df-ov 7370	. . . . . . . . . . 11 ⊢ (𝑥𝐽𝑦) = (𝐽‘⟨𝑥, 𝑦⟩)
79	77, 78	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽‘𝑧) = (𝑥𝐽𝑦))
80	76, 79	sseq12d 3955	. . . . . . . . 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 5796	. . . . . . 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 3596	. . . 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 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889  𝒫 cpw 4541  ⟨cop 4573   class class class wbr 5085   × cxp 5629  dom cdm 5631   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367  Xcixp 8845   ⊆_cat cssc 17774

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-ixp 8846  df-ssc 17777

This theorem is referenced by:  ssc1  17788  ssc2  17789  sscres  17790  ssctr  17792  0ssc  17804  catsubcat  17806  rnghmsscmap2  20606  rnghmsscmap  20607  rhmsscmap2  20635  rhmsscmap  20636  rhmsscrnghm  20642  srhmsubc  20657  fldhmsubc  20762  srhmsubcALTV  48801  fldhmsubcALTV  48809  iinfssc  49532  discsubc  49539  nelsubclem  49542  imassc  49628  setc1onsubc  50077