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Theorem isssc 17195
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 17189 . . . 4 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
2 fndm 6440 . . . . . . . . . . . 12 (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
32adantl 485 . . . . . . . . . . 11 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
4 isssc.2 . . . . . . . . . . . . 13 (𝜑𝐽 Fn (𝑇 × 𝑇))
54adantr 484 . . . . . . . . . . . 12 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
65fndmd 6442 . . . . . . . . . . 11 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑇 × 𝑇))
73, 6eqtr3d 2775 . . . . . . . . . 10 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
87dmeqd 5748 . . . . . . . . 9 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom (𝑡 × 𝑡) = dom (𝑇 × 𝑇))
9 dmxpid 5773 . . . . . . . . 9 dom (𝑡 × 𝑡) = 𝑡
10 dmxpid 5773 . . . . . . . . 9 dom (𝑇 × 𝑇) = 𝑇
118, 9, 103eqtr3g 2796 . . . . . . . 8 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
1211ex 416 . . . . . . 7 (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝑡 = 𝑇))
13 id 22 . . . . . . . . . 10 (𝑡 = 𝑇𝑡 = 𝑇)
1413sqxpeqd 5557 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡 × 𝑡) = (𝑇 × 𝑇))
1514fneq2d 6432 . . . . . . . 8 (𝑡 = 𝑇 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇)))
164, 15syl5ibrcom 250 . . . . . . 7 (𝜑 → (𝑡 = 𝑇𝐽 Fn (𝑡 × 𝑡)))
1712, 16impbid 215 . . . . . 6 (𝜑 → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝑡 = 𝑇))
1817anbi1d 633 . . . . 5 (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ (𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧))))
1918exbidv 1928 . . . 4 (𝜑 → (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ ∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧))))
201, 19syl5bb 286 . . 3 (𝜑 → (𝐻cat 𝐽 ↔ ∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧))))
21 isssc.3 . . . 4 (𝜑𝑇𝑉)
22 pweq 4504 . . . . . 6 (𝑡 = 𝑇 → 𝒫 𝑡 = 𝒫 𝑇)
2322rexeqdv 3317 . . . . 5 (𝑡 = 𝑇 → (∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
2423ceqsexgv 3550 . . . 4 (𝑇𝑉 → (∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
2521, 24syl 17 . . 3 (𝜑 → (∃𝑡(𝑡 = 𝑇 ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
2620, 25bitrd 282 . 2 (𝜑 → (𝐻cat 𝐽 ↔ ∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
27 df-rex 3059 . . 3 (∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)))
28 3anass 1096 . . . . . . . 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))))
29 elixp2 8511 . . . . . . . 8 (𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ (𝐻 ∈ V ∧ 𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))
30 vex 3402 . . . . . . . . . . . 12 𝑠 ∈ V
3130, 30xpex 7494 . . . . . . . . . . 11 (𝑠 × 𝑠) ∈ V
32 fnex 6990 . . . . . . . . . . 11 ((𝐻 Fn (𝑠 × 𝑠) ∧ (𝑠 × 𝑠) ∈ V) → 𝐻 ∈ V)
3331, 32mpan2 691 . . . . . . . . . 10 (𝐻 Fn (𝑠 × 𝑠) → 𝐻 ∈ V)
3433adantr 484 . . . . . . . . 9 ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)) → 𝐻 ∈ V)
3534pm4.71ri 564 . . . . . . . 8 ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)) ↔ (𝐻 ∈ V ∧ (𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))))
3628, 29, 353bitr4i 306 . . . . . . 7 (𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ (𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))
37 fndm 6440 . . . . . . . . . . . . . 14 (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
3837adantl 485 . . . . . . . . . . . . 13 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
39 isssc.1 . . . . . . . . . . . . . . 15 (𝜑𝐻 Fn (𝑆 × 𝑆))
4039adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
4140fndmd 6442 . . . . . . . . . . . . 13 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑆 × 𝑆))
4238, 41eqtr3d 2775 . . . . . . . . . . . 12 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
4342dmeqd 5748 . . . . . . . . . . 11 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom (𝑠 × 𝑠) = dom (𝑆 × 𝑆))
44 dmxpid 5773 . . . . . . . . . . 11 dom (𝑠 × 𝑠) = 𝑠
45 dmxpid 5773 . . . . . . . . . . 11 dom (𝑆 × 𝑆) = 𝑆
4643, 44, 453eqtr3g 2796 . . . . . . . . . 10 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
4746ex 416 . . . . . . . . 9 (𝜑 → (𝐻 Fn (𝑠 × 𝑠) → 𝑠 = 𝑆))
48 id 22 . . . . . . . . . . . 12 (𝑠 = 𝑆𝑠 = 𝑆)
4948sqxpeqd 5557 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑠 × 𝑠) = (𝑆 × 𝑆))
5049fneq2d 6432 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝐻 Fn (𝑆 × 𝑆)))
5139, 50syl5ibrcom 250 . . . . . . . . 9 (𝜑 → (𝑠 = 𝑆𝐻 Fn (𝑠 × 𝑠)))
5247, 51impbid 215 . . . . . . . 8 (𝜑 → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝑠 = 𝑆))
5352anbi1d 633 . . . . . . 7 (𝜑 → ((𝐻 Fn (𝑠 × 𝑠) ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)) ↔ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))))
5436, 53syl5bb 286 . . . . . 6 (𝜑 → (𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))))
5554anbi2d 632 . . . . 5 (𝜑 → ((𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ (𝑠 ∈ 𝒫 𝑇 ∧ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))))
56 an12 645 . . . . 5 ((𝑠 ∈ 𝒫 𝑇 ∧ (𝑠 = 𝑆 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) ↔ (𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))))
5755, 56bitrdi 290 . . . 4 (𝜑 → ((𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ (𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))))
5857exbidv 1928 . . 3 (𝜑 → (∃𝑠(𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧)) ↔ ∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))))
5927, 58syl5bb 286 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝑇𝐻X𝑧 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑧) ↔ ∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))))
60 exsimpl 1875 . . . . 5 (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) → ∃𝑠 𝑠 = 𝑆)
61 isset 3411 . . . . 5 (𝑆 ∈ V ↔ ∃𝑠 𝑠 = 𝑆)
6260, 61sylibr 237 . . . 4 (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) → 𝑆 ∈ V)
6362a1i 11 . . 3 (𝜑 → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) → 𝑆 ∈ V))
64 ssexg 5191 . . . . . 6 ((𝑆𝑇𝑇𝑉) → 𝑆 ∈ V)
6564expcom 417 . . . . 5 (𝑇𝑉 → (𝑆𝑇𝑆 ∈ V))
6621, 65syl 17 . . . 4 (𝜑 → (𝑆𝑇𝑆 ∈ V))
6766adantrd 495 . . 3 (𝜑 → ((𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → 𝑆 ∈ V))
6830elpw 4492 . . . . . . 7 (𝑠 ∈ 𝒫 𝑇𝑠𝑇)
69 sseq1 3902 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑇𝑆𝑇))
7068, 69syl5bb 286 . . . . . 6 (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 𝑇𝑆𝑇))
7149raleqdv 3316 . . . . . . 7 (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)))
72 fvex 6687 . . . . . . . . . 10 (𝐻𝑧) ∈ V
7372elpw 4492 . . . . . . . . 9 ((𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ (𝐻𝑧) ⊆ (𝐽𝑧))
74 fveq2 6674 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
75 df-ov 7173 . . . . . . . . . . 11 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
7674, 75eqtr4di 2791 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
77 fveq2 6674 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽𝑧) = (𝐽‘⟨𝑥, 𝑦⟩))
78 df-ov 7173 . . . . . . . . . . 11 (𝑥𝐽𝑦) = (𝐽‘⟨𝑥, 𝑦⟩)
7977, 78eqtr4di 2791 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐽𝑧) = (𝑥𝐽𝑦))
8076, 79sseq12d 3910 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻𝑧) ⊆ (𝐽𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
8173, 80syl5bb 286 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
8281ralxp 5684 . . . . . . 7 (∀𝑧 ∈ (𝑆 × 𝑆)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
8371, 82bitrdi 290 . . . . . 6 (𝑠 = 𝑆 → (∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
8470, 83anbi12d 634 . . . . 5 (𝑠 = 𝑆 → ((𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧)) ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
8584ceqsexgv 3550 . . . 4 (𝑆 ∈ V → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
8685a1i 11 . . 3 (𝜑 → (𝑆 ∈ V → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))))
8763, 67, 86pm5.21ndd 384 . 2 (𝜑 → (∃𝑠(𝑠 = 𝑆 ∧ (𝑠 ∈ 𝒫 𝑇 ∧ ∀𝑧 ∈ (𝑠 × 𝑠)(𝐻𝑧) ∈ 𝒫 (𝐽𝑧))) ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
8826, 59, 873bitrd 308 1 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  wral 3053  wrex 3054  Vcvv 3398  wss 3843  𝒫 cpw 4488  cop 4522   class class class wbr 5030   × cxp 5523  dom cdm 5525   Fn wfn 6334  cfv 6339  (class class class)co 7170  Xcixp 8507  cat cssc 17182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-ixp 8508  df-ssc 17185
This theorem is referenced by:  ssc1  17196  ssc2  17197  sscres  17198  ssctr  17200  0ssc  17212  catsubcat  17214  rnghmsscmap2  45065  rnghmsscmap  45066  rhmsscmap2  45111  rhmsscmap  45112  rhmsscrnghm  45118  srhmsubc  45168  fldhmsubc  45176  srhmsubcALTV  45186  fldhmsubcALTV  45194
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