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Theorem domss2 8070
Description: A corollary of disjenex 8069. 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 6110 . . . . . . . 8 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
213ad2ant1 1080 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹:𝐴1-1-onto→ran 𝐹)
3 simp2 1060 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐴𝑉)
4 rnexg 7052 . . . . . . . . . 10 (𝐴𝑉 → ran 𝐴 ∈ V)
53, 4syl 17 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ran 𝐴 ∈ V)
6 uniexg 6915 . . . . . . . . 9 (ran 𝐴 ∈ V → ran 𝐴 ∈ V)
7 pwexg 4815 . . . . . . . . 9 ( ran 𝐴 ∈ V → 𝒫 ran 𝐴 ∈ V)
85, 6, 73syl 18 . . . . . . . 8 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝒫 ran 𝐴 ∈ V)
9 1stconst 7217 . . . . . . . 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 4773 . . . . . . . . . 10 (𝐵𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
12113ad2ant3 1082 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
13 disjen 8068 . . . . . . . . 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 692 . . . . . . . 8 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) = ∅ ∧ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ≈ (𝐵 ∖ ran 𝐹)))
1514simpld 475 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) = ∅)
16 disjdif 4017 . . . . . . . 8 (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅
1716a1i 11 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅)
18 f1oun 6118 . . . . . . 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 1324 . . . . . 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 4021 . . . . . . . 8 (ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) = (ran 𝐹𝐵)
21 f1f 6063 . . . . . . . . . . 11 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
22213ad2ant1 1080 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹:𝐴𝐵)
23 frn 6015 . . . . . . . . . 10 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
2422, 23syl 17 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ran 𝐹𝐵)
25 ssequn1 3766 . . . . . . . . 9 (ran 𝐹𝐵 ↔ (ran 𝐹𝐵) = 𝐵)
2624, 25sylib 208 . . . . . . . 8 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹𝐵) = 𝐵)
2720, 26syl5eq 2667 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) = 𝐵)
28 f1oeq3 6091 . . . . . . 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 6111 . . . . 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 6089 . . . . 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 6106 . . . . 5 (𝐺:𝐵1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) → 𝐺:𝐵onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
38 forn 6080 . . . . 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 6091 . . . 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 3759 . . 3 𝐴 ⊆ (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))
4443, 39syl5sseqr 3638 . 2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐴 ⊆ ran 𝐺)
45 ssid 3608 . . . 4 ran 𝐹 ⊆ ran 𝐹
46 cores 5602 . . . 4 (ran 𝐹 ⊆ ran 𝐹 → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐺𝐹))
4745, 46ax-mp 5 . . 3 ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐺𝐹)
48 dmres 5383 . . . . . . . . 9 dom ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) = (ran 𝐹 ∩ dom (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
49 f1ocnv 6111 . . . . . . . . . . . 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 6103 . . . . . . . . . . . 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 3797 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹 ∩ dom (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)))
5352, 16syl6eq 2671 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (ran 𝐹 ∩ dom (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = ∅)
5448, 53syl5eq 2667 . . . . . . . 8 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → dom ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹) = ∅)
55 relres 5390 . . . . . . . . 9 Rel ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹)
56 reldm0 5308 . . . . . . . . 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 3750 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹)) = (𝐹 ∪ ∅))
60 cnvun 5502 . . . . . . . . 9 (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = (𝐹(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
6133, 60eqtri 2643 . . . . . . . 8 𝐺 = (𝐹(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
6261reseq1i 5357 . . . . . . 7 (𝐺 ↾ ran 𝐹) = ((𝐹(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ↾ ran 𝐹)
63 resundir 5375 . . . . . . 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 5090 . . . . . . . . . 10 ran 𝐹 = dom 𝐹
6564reseq2i 5358 . . . . . . . . 9 (𝐹 ↾ ran 𝐹) = (𝐹 ↾ dom 𝐹)
66 relcnv 5467 . . . . . . . . . 10 Rel 𝐹
67 resdm 5405 . . . . . . . . . 10 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
6866, 67ax-mp 5 . . . . . . . . 9 (𝐹 ↾ dom 𝐹) = 𝐹
6965, 68eqtri 2643 . . . . . . . 8 (𝐹 ↾ ran 𝐹) = 𝐹
7069uneq1i 3746 . . . . . . 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 2648 . . . . . 6 (𝐹 ∪ ((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ↾ ran 𝐹)) = (𝐺 ↾ ran 𝐹)
72 un0 3944 . . . . . 6 (𝐹 ∪ ∅) = 𝐹
7359, 71, 723eqtr3g 2678 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐺 ↾ ran 𝐹) = 𝐹)
7473coeq1d 5248 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐹𝐹))
75 f1cocnv1 6128 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
76753ad2ant1 1080 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹𝐹) = ( I ↾ 𝐴))
7774, 76eqtrd 2655 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = ( I ↾ 𝐴))
7847, 77syl5eqr 2669 . 2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐺𝐹) = ( I ↾ 𝐴))
7942, 44, 783jca 1240 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 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407   class class class wbr 4618   I cid 4989   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  ccom 5083  Rel wrel 5084  wf 5848  1-1wf1 5849  ontowfo 5850  1-1-ontowf1o 5851  1st c1st 7118  cen 7903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-1st 7120  df-2nd 7121  df-en 7907
This theorem is referenced by:  domssex2  8071  domssex  8072
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