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Theorem isssc 16680
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 (𝜑𝐻 Fn (𝑆 × 𝑆))
isssc.2 (𝜑𝐽 Fn (𝑇 × 𝑇))
isssc.3 (𝜑𝑇𝑉)
Assertion
Ref Expression
isssc (𝜑 → (𝐻cat 𝐽 ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (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.
StepHypRef Expression
1 brssc 16674 . . . 4 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
2 fndm 6128 . . . . . . . . . . . 12 (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
32adantl 467 . . . . . . . . . . 11 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
4 isssc.2 . . . . . . . . . . . . 13 (𝜑𝐽 Fn (𝑇 × 𝑇))
54adantr 466 . . . . . . . . . . . 12 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
6 fndm 6128 . . . . . . . . . . . 12 (𝐽 Fn (𝑇 × 𝑇) → dom 𝐽 = (𝑇 × 𝑇))
75, 6syl 17 . . . . . . . . . . 11 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑇 × 𝑇))
83, 7eqtr3d 2807 . . . . . . . . . 10 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
98dmeqd 5462 . . . . . . . . 9 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom (𝑡 × 𝑡) = dom (𝑇 × 𝑇))
10 dmxpid 5481 . . . . . . . . 9 dom (𝑡 × 𝑡) = 𝑡
11 dmxpid 5481 . . . . . . . . 9 dom (𝑇 × 𝑇) = 𝑇
129, 10, 113eqtr3g 2828 . . . . . . . 8 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
1312ex 397 . . . . . . 7 (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝑡 = 𝑇))
14 id 22 . . . . . . . . . 10 (𝑡 = 𝑇𝑡 = 𝑇)
1514sqxpeqd 5280 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡 × 𝑡) = (𝑇 × 𝑇))
1615fneq2d 6120 . . . . . . . 8 (𝑡 = 𝑇 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇)))
174, 16syl5ibrcom 237 . . . . . . 7 (𝜑 → (𝑡 = 𝑇𝐽 Fn (𝑡 × 𝑡)))
1813, 17impbid 202 . . . . . 6 (𝜑 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝑡 = 𝑇))
1918anbi1d 615 . . . . 5 (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ (𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧))))
2019exbidv 2002 . . . 4 (𝜑 → (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ ∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧))))
211, 20syl5bb 272 . . 3 (𝜑 → (𝐻cat 𝐽 ↔ ∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧))))
22 isssc.3 . . . 4 (𝜑𝑇𝑉)
23 pweq 4300 . . . . . 6 (𝑡 = 𝑇 → 𝒫 𝑡 = 𝒫 𝑇)
2423rexeqdv 3294 . . . . 5 (𝑡 = 𝑇 → (∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
2524ceqsexgv 3485 . . . 4 (𝑇𝑉 → (∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
2622, 25syl 17 . . 3 (𝜑 → (∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
2721, 26bitrd 268 . 2 (𝜑 → (𝐻cat 𝐽 ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
28 df-rex 3067 . . 3 (∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
29 3anass 1080 . . . . . . . 8 ((𝐻 ∈ V ∧ 𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)) ↔ (𝐻 ∈ V ∧ (𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))))
30 elixp2 8064 . . . . . . . 8 (𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ (𝐻 ∈ V ∧ 𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))
31 vex 3354 . . . . . . . . . . . 12 𝑠 ∈ V
3231, 31xpex 7107 . . . . . . . . . . 11 (𝑠 × 𝑠) ∈ V
33 fnex 6623 . . . . . . . . . . 11 ((𝐻 Fn (𝑠 × 𝑠) ∧ (𝑠 × 𝑠) ∈ V) → 𝐻 ∈ V)
3432, 33mpan2 671 . . . . . . . . . 10 (𝐻 Fn (𝑠 × 𝑠) → 𝐻 ∈ V)
3534adantr 466 . . . . . . . . 9 ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)) → 𝐻 ∈ V)
3635pm4.71ri 550 . . . . . . . 8 ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)) ↔ (𝐻 ∈ V ∧ (𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))))
3729, 30, 363bitr4i 292 . . . . . . 7 (𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ (𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))
38 fndm 6128 . . . . . . . . . . . . . 14 (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
3938adantl 467 . . . . . . . . . . . . 13 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
40 isssc.1 . . . . . . . . . . . . . . 15 (𝜑𝐻 Fn (𝑆 × 𝑆))
4140adantr 466 . . . . . . . . . . . . . 14 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
42 fndm 6128 . . . . . . . . . . . . . 14 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
4341, 42syl 17 . . . . . . . . . . . . 13 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑆 × 𝑆))
4439, 43eqtr3d 2807 . . . . . . . . . . . 12 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
4544dmeqd 5462 . . . . . . . . . . 11 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom (𝑠 × 𝑠) = dom (𝑆 × 𝑆))
46 dmxpid 5481 . . . . . . . . . . 11 dom (𝑠 × 𝑠) = 𝑠
47 dmxpid 5481 . . . . . . . . . . 11 dom (𝑆 × 𝑆) = 𝑆
4845, 46, 473eqtr3g 2828 . . . . . . . . . 10 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
4948ex 397 . . . . . . . . 9 (𝜑 → (𝐻 Fn (𝑠 × 𝑠) → 𝑠 = 𝑆))
50 id 22 . . . . . . . . . . . 12 (𝑠 = 𝑆𝑠 = 𝑆)
5150sqxpeqd 5280 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑠 × 𝑠) = (𝑆 × 𝑆))
5251fneq2d 6120 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝐻 Fn (𝑆 × 𝑆)))
5340, 52syl5ibrcom 237 . . . . . . . . 9 (𝜑 → (𝑠 = 𝑆𝐻 Fn (𝑠 × 𝑠)))
5449, 53impbid 202 . . . . . . . 8 (𝜑 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝑠 = 𝑆))
5554anbi1d 615 . . . . . . 7 (𝜑 → ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)) ↔ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))))
5637, 55syl5bb 272 . . . . . 6 (𝜑 → (𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))))
5756anbi2d 614 . . . . 5 (𝜑 → ((𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ (𝑠 ∈ 𝒫 𝑇 ∧ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))))
58 an12 624 . . . . 5 ((𝑠 ∈ 𝒫 𝑇 ∧ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) ↔ (𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))))
5957, 58syl6bb 276 . . . 4 (𝜑 → ((𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ (𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))))
6059exbidv 2002 . . 3 (𝜑 → (∃𝑠(𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ ∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))))
6128, 60syl5bb 272 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ ∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))))
62 exsimpl 1946 . . . . 5 (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) → ∃𝑠 𝑠 = 𝑆)
63 isset 3359 . . . . 5 (𝑆 ∈ V ↔ ∃𝑠 𝑠 = 𝑆)
6462, 63sylibr 224 . . . 4 (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) → 𝑆 ∈ V)
6564a1i 11 . . 3 (𝜑 → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) → 𝑆 ∈ V))
66 ssexg 4938 . . . . . 6 ((𝑆𝑇𝑇𝑉) → 𝑆 ∈ V)
6766expcom 398 . . . . 5 (𝑇𝑉 → (𝑆𝑇𝑆 ∈ V))
6822, 67syl 17 . . . 4 (𝜑 → (𝑆𝑇𝑆 ∈ V))
6968adantrd 479 . . 3 (𝜑 → ((𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → 𝑆 ∈ V))
7031elpw 4303 . . . . . . 7 (𝑠 ∈ 𝒫 𝑇𝑠𝑇)
71 sseq1 3775 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑇𝑆𝑇))
7270, 71syl5bb 272 . . . . . 6 (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 𝑇𝑆𝑇))
7351raleqdv 3293 . . . . . . 7 (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))
74 fvex 6340 . . . . . . . . . 10 (𝐻𝑧) ∈ V
7574elpw 4303 . . . . . . . . 9 ((𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ (𝐻𝑧) ⊆ (𝐽𝑧))
76 fveq2 6330 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
77 df-ov 6794 . . . . . . . . . . 11 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
7876, 77syl6eqr 2823 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
79 fveq2 6330 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽𝑧) = (𝐽‘⟨𝑥, 𝑦⟩))
80 df-ov 6794 . . . . . . . . . . 11 (𝑥𝐽𝑦) = (𝐽‘⟨𝑥, 𝑦⟩)
8179, 80syl6eqr 2823 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽𝑧) = (𝑥𝐽𝑦))
8278, 81sseq12d 3783 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻𝑧) ⊆ (𝐽𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
8375, 82syl5bb 272 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
8483ralxp 5400 . . . . . . 7 (∀𝑧 ∈ (𝑆 × 𝑆)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
8573, 84syl6bb 276 . . . . . 6 (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
8672, 85anbi12d 616 . . . . 5 (𝑠 = 𝑆 → ((𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)) ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
8786ceqsexgv 3485 . . . 4 (𝑆 ∈ V → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
8887a1i 11 . . 3 (𝜑 → (𝑆 ∈ V → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))))
8965, 69, 88pm5.21ndd 368 . 2 (𝜑 → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
9027, 61, 893bitrd 294 1 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  wss 3723  𝒫 cpw 4297  cop 4322   class class class wbr 4786   × cxp 5247  dom cdm 5249   Fn wfn 6024  cfv 6029  (class class class)co 6791  Xcixp 8060  cat cssc 16667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6794  df-ixp 8061  df-ssc 16670
This theorem is referenced by:  ssc1  16681  ssc2  16682  sscres  16683  ssctr  16685  0ssc  16697  catsubcat  16699  rnghmsscmap2  42494  rnghmsscmap  42495  rhmsscmap2  42540  rhmsscmap  42541  rhmsscrnghm  42547  srhmsubc  42597  fldhmsubc  42605  srhmsubcALTV  42615  fldhmsubcALTV  42623
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