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Theorem domss2 8273
Description: A corollary of disjenex 8272. If 𝐹 is an injection from 𝐴 to 𝐵 then 𝐺 is a right inverse of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
domss2.1 𝐺 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
Assertion
Ref Expression
domss2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐺:𝐵1-1-onto→ran 𝐺𝐴 ⊆ ran 𝐺 ∧ (𝐺𝐹) = ( I ↾ 𝐴)))

Proof of Theorem domss2
StepHypRef Expression
1 f1f1orn 6287 . . . . . . . 8 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
213ad2ant1 1127 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹:𝐴1-1-onto→ran 𝐹)
3 simp2 1131 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐴𝑉)
4 rnexg 7243 . . . . . . . . . 10 (𝐴𝑉 → ran 𝐴 ∈ V)
53, 4syl 17 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ran 𝐴 ∈ V)
6 uniexg 7100 . . . . . . . . 9 (ran 𝐴 ∈ V → ran 𝐴 ∈ V)
7 pwexg 4980 . . . . . . . . 9 ( ran 𝐴 ∈ V → 𝒫 ran 𝐴 ∈ V)
85, 6, 73syl 18 . . . . . . . 8 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝒫 ran 𝐴 ∈ V)
9 1stconst 7414 . . . . . . . 8 (𝒫 ran 𝐴 ∈ V → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹))
108, 9syl 17 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹))
11 difexg 4942 . . . . . . . . . 10 (𝐵𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
12113ad2ant3 1129 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
13 disjen 8271 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝐵 ∖ ran 𝐹) ∈ V) → ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) = ∅ ∧ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ≈ (𝐵 ∖ ran 𝐹)))
143, 12, 13syl2anc 573 . . . . . . . 8 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) = ∅ ∧ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ≈ (𝐵 ∖ ran 𝐹)))
1514simpld 482 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) = ∅)
16 disjdif 4182 . . . . . . . 8 (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅
1716a1i 11 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅)
18 f1oun 6295 . . . . . . 7 (((𝐹:𝐴1-1-onto→ran 𝐹 ∧ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹)) ∧ ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) = ∅ ∧ (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅)) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))–1-1-onto→(ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)))
192, 10, 15, 17, 18syl22anc 1477 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))–1-1-onto→(ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)))
20 undif2 4186 . . . . . . . 8 (ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) = (ran 𝐹𝐵)
21 f1f 6239 . . . . . . . . . . 11 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
22213ad2ant1 1127 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹:𝐴𝐵)
23 frn 6191 . . . . . . . . . 10 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
2422, 23syl 17 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ran 𝐹𝐵)
25 ssequn1 3934 . . . . . . . . 9 (ran 𝐹𝐵 ↔ (ran 𝐹𝐵) = 𝐵)
2624, 25sylib 208 . . . . . . . 8 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹𝐵) = 𝐵)
2720, 26syl5eq 2817 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) = 𝐵)
28 f1oeq3 6268 . . . . . . 7 ((ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) = 𝐵 → ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))–1-1-onto→(ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) ↔ (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))–1-1-onto𝐵))
2927, 28syl 17 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))–1-1-onto→(ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) ↔ (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))–1-1-onto𝐵))
3019, 29mpbid 222 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))–1-1-onto𝐵)
31 f1ocnv 6288 . . . . 5 ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))–1-1-onto𝐵(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
3230, 31syl 17 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
33 domss2.1 . . . . 5 𝐺 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
34 f1oeq1 6266 . . . . 5 (𝐺 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → (𝐺:𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↔ (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))))
3533, 34ax-mp 5 . . . 4 (𝐺:𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↔ (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
3632, 35sylibr 224 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐺:𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
37 f1ofo 6283 . . . . 5 (𝐺:𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) → 𝐺:𝐵onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
38 forn 6257 . . . . 5 (𝐺:𝐵onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) → ran 𝐺 = (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
3936, 37, 383syl 18 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ran 𝐺 = (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
40 f1oeq3 6268 . . . 4 (ran 𝐺 = (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) → (𝐺:𝐵1-1-onto→ran 𝐺𝐺:𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))))
4139, 40syl 17 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐺:𝐵1-1-onto→ran 𝐺𝐺:𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))))
4236, 41mpbird 247 . 2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐺:𝐵1-1-onto→ran 𝐺)
43 ssun1 3927 . . 3 𝐴 ⊆ (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))
4443, 39syl5sseqr 3803 . 2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐴 ⊆ ran 𝐺)
45 ssid 3773 . . . 4 ran 𝐹 ⊆ ran 𝐹
46 cores 5780 . . . 4 (ran 𝐹 ⊆ ran 𝐹 → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐺𝐹))
4745, 46ax-mp 5 . . 3 ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐺𝐹)
48 dmres 5558 . . . . . . . . 9 dom ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) = (ran 𝐹 ∩ dom (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
49 f1ocnv 6288 . . . . . . . . . . . 12 ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹) → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):(𝐵 ∖ ran 𝐹)–1-1-onto→((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))
50 f1odm 6280 . . . . . . . . . . . 12 ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):(𝐵 ∖ ran 𝐹)–1-1-onto→((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) → dom (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) = (𝐵 ∖ ran 𝐹))
5110, 49, 503syl 18 . . . . . . . . . . 11 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → dom (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) = (𝐵 ∖ ran 𝐹))
5251ineq2d 3965 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹 ∩ dom (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)))
5352, 16syl6eq 2821 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹 ∩ dom (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = ∅)
5448, 53syl5eq 2817 . . . . . . . 8 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → dom ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) = ∅)
55 relres 5565 . . . . . . . . 9 Rel ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹)
56 reldm0 5479 . . . . . . . . 9 (Rel ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) → (((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) = ∅ ↔ dom ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) = ∅))
5755, 56ax-mp 5 . . . . . . . 8 (((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) = ∅ ↔ dom ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) = ∅)
5854, 57sylibr 224 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) = ∅)
5958uneq2d 3918 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹)) = (𝐹 ∪ ∅))
60 cnvun 5677 . . . . . . . . 9 (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = (𝐹(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
6133, 60eqtri 2793 . . . . . . . 8 𝐺 = (𝐹(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
6261reseq1i 5528 . . . . . . 7 (𝐺 ↾ ran 𝐹) = ((𝐹(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ↾ ran 𝐹)
63 resundir 5550 . . . . . . 7 ((𝐹(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ↾ ran 𝐹) = ((𝐹 ↾ ran 𝐹) ∪ ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹))
64 df-rn 5260 . . . . . . . . . 10 ran 𝐹 = dom 𝐹
6564reseq2i 5529 . . . . . . . . 9 (𝐹 ↾ ran 𝐹) = (𝐹 ↾ dom 𝐹)
66 relcnv 5642 . . . . . . . . . 10 Rel 𝐹
67 resdm 5580 . . . . . . . . . 10 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
6866, 67ax-mp 5 . . . . . . . . 9 (𝐹 ↾ dom 𝐹) = 𝐹
6965, 68eqtri 2793 . . . . . . . 8 (𝐹 ↾ ran 𝐹) = 𝐹
7069uneq1i 3914 . . . . . . 7 ((𝐹 ↾ ran 𝐹) ∪ ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹)) = (𝐹 ∪ ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹))
7162, 63, 703eqtrri 2798 . . . . . 6 (𝐹 ∪ ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹)) = (𝐺 ↾ ran 𝐹)
72 un0 4111 . . . . . 6 (𝐹 ∪ ∅) = 𝐹
7359, 71, 723eqtr3g 2828 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐺 ↾ ran 𝐹) = 𝐹)
7473coeq1d 5420 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐹𝐹))
75 f1cocnv1 6305 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
76753ad2ant1 1127 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹𝐹) = ( I ↾ 𝐴))
7774, 76eqtrd 2805 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = ( I ↾ 𝐴))
7847, 77syl5eqr 2819 . 2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐺𝐹) = ( I ↾ 𝐴))
7942, 44, 783jca 1122 1 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐺:𝐵1-1-onto→ran 𝐺𝐴 ⊆ ran 𝐺 ∧ (𝐺𝐹) = ( I ↾ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316   cuni 4574   class class class wbr 4786   I cid 5156   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  ccom 5253  Rel wrel 5254  wf 6025  1-1wf1 6026  ontowfo 6027  1-1-ontowf1o 6028  1st c1st 7311  cen 8104
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-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-nel 3047  df-ral 3066  df-rex 3067  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-int 4612  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-1st 7313  df-2nd 7314  df-en 8108
This theorem is referenced by:  domssex2  8274  domssex  8275
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