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Theorem ssenen 8341
Description: Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
ssenen (𝐴𝐵 → {𝑥 ∣ (𝑥𝐴𝑥𝐶)} ≈ {𝑥 ∣ (𝑥𝐵𝑥𝐶)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem ssenen
Dummy variables 𝑦 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bren 8169 . . 3 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
2 f1odm 6324 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → dom 𝑓 = 𝐴)
3 vex 3353 . . . . . . . 8 𝑓 ∈ V
43dmex 7297 . . . . . . 7 dom 𝑓 ∈ V
52, 4syl6eqelr 2853 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝐴 ∈ V)
6 pwexg 5014 . . . . . 6 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
7 inex1g 4962 . . . . . 6 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∈ V)
85, 6, 73syl 18 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∈ V)
9 f1ofo 6327 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
10 forn 6301 . . . . . . . 8 (𝑓:𝐴onto𝐵 → ran 𝑓 = 𝐵)
119, 10syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → ran 𝑓 = 𝐵)
123rnex 7298 . . . . . . 7 ran 𝑓 ∈ V
1311, 12syl6eqelr 2853 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝐵 ∈ V)
14 pwexg 5014 . . . . . 6 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
15 inex1g 4962 . . . . . 6 (𝒫 𝐵 ∈ V → (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ∈ V)
1613, 14, 153syl 18 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ∈ V)
17 f1of1 6319 . . . . . . . . . . 11 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴1-1𝐵)
1817adantr 472 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → 𝑓:𝐴1-1𝐵)
1913adantr 472 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → 𝐵 ∈ V)
20 simpr 477 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → 𝑦𝐴)
21 vex 3353 . . . . . . . . . . 11 𝑦 ∈ V
2221a1i 11 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → 𝑦 ∈ V)
23 f1imaen2g 8221 . . . . . . . . . 10 (((𝑓:𝐴1-1𝐵𝐵 ∈ V) ∧ (𝑦𝐴𝑦 ∈ V)) → (𝑓𝑦) ≈ 𝑦)
2418, 19, 20, 22, 23syl22anc 867 . . . . . . . . 9 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → (𝑓𝑦) ≈ 𝑦)
25 entr 8212 . . . . . . . . 9 (((𝑓𝑦) ≈ 𝑦𝑦𝐶) → (𝑓𝑦) ≈ 𝐶)
2624, 25sylan 575 . . . . . . . 8 (((𝑓:𝐴1-1-onto𝐵𝑦𝐴) ∧ 𝑦𝐶) → (𝑓𝑦) ≈ 𝐶)
2726expl 449 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → ((𝑦𝐴𝑦𝐶) → (𝑓𝑦) ≈ 𝐶))
28 imassrn 5659 . . . . . . . . 9 (𝑓𝑦) ⊆ ran 𝑓
2928, 10syl5sseq 3813 . . . . . . . 8 (𝑓:𝐴onto𝐵 → (𝑓𝑦) ⊆ 𝐵)
309, 29syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓𝑦) ⊆ 𝐵)
3127, 30jctild 521 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ((𝑦𝐴𝑦𝐶) → ((𝑓𝑦) ⊆ 𝐵 ∧ (𝑓𝑦) ≈ 𝐶)))
32 elin 3958 . . . . . . 7 (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ↔ (𝑦 ∈ 𝒫 𝐴𝑦 ∈ {𝑥𝑥𝐶}))
3321elpw 4321 . . . . . . . 8 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
34 breq1 4812 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐶𝑦𝐶))
3521, 34elab 3504 . . . . . . . 8 (𝑦 ∈ {𝑥𝑥𝐶} ↔ 𝑦𝐶)
3633, 35anbi12i 620 . . . . . . 7 ((𝑦 ∈ 𝒫 𝐴𝑦 ∈ {𝑥𝑥𝐶}) ↔ (𝑦𝐴𝑦𝐶))
3732, 36bitri 266 . . . . . 6 (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ↔ (𝑦𝐴𝑦𝐶))
38 elin 3958 . . . . . . 7 ((𝑓𝑦) ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ↔ ((𝑓𝑦) ∈ 𝒫 𝐵 ∧ (𝑓𝑦) ∈ {𝑥𝑥𝐶}))
393imaex 7302 . . . . . . . . 9 (𝑓𝑦) ∈ V
4039elpw 4321 . . . . . . . 8 ((𝑓𝑦) ∈ 𝒫 𝐵 ↔ (𝑓𝑦) ⊆ 𝐵)
41 breq1 4812 . . . . . . . . 9 (𝑥 = (𝑓𝑦) → (𝑥𝐶 ↔ (𝑓𝑦) ≈ 𝐶))
4239, 41elab 3504 . . . . . . . 8 ((𝑓𝑦) ∈ {𝑥𝑥𝐶} ↔ (𝑓𝑦) ≈ 𝐶)
4340, 42anbi12i 620 . . . . . . 7 (((𝑓𝑦) ∈ 𝒫 𝐵 ∧ (𝑓𝑦) ∈ {𝑥𝑥𝐶}) ↔ ((𝑓𝑦) ⊆ 𝐵 ∧ (𝑓𝑦) ≈ 𝐶))
4438, 43bitri 266 . . . . . 6 ((𝑓𝑦) ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ↔ ((𝑓𝑦) ⊆ 𝐵 ∧ (𝑓𝑦) ≈ 𝐶))
4531, 37, 443imtr4g 287 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) → (𝑓𝑦) ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶})))
46 f1ocnv 6332 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
47 f1of1 6319 . . . . . . . . . . . 12 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵1-1𝐴)
48 f1f1orn 6331 . . . . . . . . . . . 12 (𝑓:𝐵1-1𝐴𝑓:𝐵1-1-onto→ran 𝑓)
49 f1of1 6319 . . . . . . . . . . . 12 (𝑓:𝐵1-1-onto→ran 𝑓𝑓:𝐵1-1→ran 𝑓)
5047, 48, 493syl 18 . . . . . . . . . . 11 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵1-1→ran 𝑓)
51 vex 3353 . . . . . . . . . . . 12 𝑧 ∈ V
5251f1imaen 8222 . . . . . . . . . . 11 ((𝑓:𝐵1-1→ran 𝑓𝑧𝐵) → (𝑓𝑧) ≈ 𝑧)
5350, 52sylan 575 . . . . . . . . . 10 ((𝑓:𝐵1-1-onto𝐴𝑧𝐵) → (𝑓𝑧) ≈ 𝑧)
54 entr 8212 . . . . . . . . . 10 (((𝑓𝑧) ≈ 𝑧𝑧𝐶) → (𝑓𝑧) ≈ 𝐶)
5553, 54sylan 575 . . . . . . . . 9 (((𝑓:𝐵1-1-onto𝐴𝑧𝐵) ∧ 𝑧𝐶) → (𝑓𝑧) ≈ 𝐶)
5655expl 449 . . . . . . . 8 (𝑓:𝐵1-1-onto𝐴 → ((𝑧𝐵𝑧𝐶) → (𝑓𝑧) ≈ 𝐶))
57 f1ofo 6327 . . . . . . . . 9 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵onto𝐴)
58 imassrn 5659 . . . . . . . . . 10 (𝑓𝑧) ⊆ ran 𝑓
59 forn 6301 . . . . . . . . . 10 (𝑓:𝐵onto𝐴 → ran 𝑓 = 𝐴)
6058, 59syl5sseq 3813 . . . . . . . . 9 (𝑓:𝐵onto𝐴 → (𝑓𝑧) ⊆ 𝐴)
6157, 60syl 17 . . . . . . . 8 (𝑓:𝐵1-1-onto𝐴 → (𝑓𝑧) ⊆ 𝐴)
6256, 61jctild 521 . . . . . . 7 (𝑓:𝐵1-1-onto𝐴 → ((𝑧𝐵𝑧𝐶) → ((𝑓𝑧) ⊆ 𝐴 ∧ (𝑓𝑧) ≈ 𝐶)))
6346, 62syl 17 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ((𝑧𝐵𝑧𝐶) → ((𝑓𝑧) ⊆ 𝐴 ∧ (𝑓𝑧) ≈ 𝐶)))
64 elin 3958 . . . . . . 7 (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ↔ (𝑧 ∈ 𝒫 𝐵𝑧 ∈ {𝑥𝑥𝐶}))
6551elpw 4321 . . . . . . . 8 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
66 breq1 4812 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝐶𝑧𝐶))
6751, 66elab 3504 . . . . . . . 8 (𝑧 ∈ {𝑥𝑥𝐶} ↔ 𝑧𝐶)
6865, 67anbi12i 620 . . . . . . 7 ((𝑧 ∈ 𝒫 𝐵𝑧 ∈ {𝑥𝑥𝐶}) ↔ (𝑧𝐵𝑧𝐶))
6964, 68bitri 266 . . . . . 6 (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ↔ (𝑧𝐵𝑧𝐶))
70 elin 3958 . . . . . . 7 ((𝑓𝑧) ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ↔ ((𝑓𝑧) ∈ 𝒫 𝐴 ∧ (𝑓𝑧) ∈ {𝑥𝑥𝐶}))
713cnvex 7311 . . . . . . . . . 10 𝑓 ∈ V
7271imaex 7302 . . . . . . . . 9 (𝑓𝑧) ∈ V
7372elpw 4321 . . . . . . . 8 ((𝑓𝑧) ∈ 𝒫 𝐴 ↔ (𝑓𝑧) ⊆ 𝐴)
74 breq1 4812 . . . . . . . . 9 (𝑥 = (𝑓𝑧) → (𝑥𝐶 ↔ (𝑓𝑧) ≈ 𝐶))
7572, 74elab 3504 . . . . . . . 8 ((𝑓𝑧) ∈ {𝑥𝑥𝐶} ↔ (𝑓𝑧) ≈ 𝐶)
7673, 75anbi12i 620 . . . . . . 7 (((𝑓𝑧) ∈ 𝒫 𝐴 ∧ (𝑓𝑧) ∈ {𝑥𝑥𝐶}) ↔ ((𝑓𝑧) ⊆ 𝐴 ∧ (𝑓𝑧) ≈ 𝐶))
7770, 76bitri 266 . . . . . 6 ((𝑓𝑧) ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ↔ ((𝑓𝑧) ⊆ 𝐴 ∧ (𝑓𝑧) ≈ 𝐶))
7863, 69, 773imtr4g 287 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) → (𝑓𝑧) ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶})))
79 simpl 474 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 𝐵𝑧 ∈ {𝑥𝑥𝐶}) → 𝑧 ∈ 𝒫 𝐵)
8079elpwid 4327 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 𝐵𝑧 ∈ {𝑥𝑥𝐶}) → 𝑧𝐵)
8164, 80sylbi 208 . . . . . . . . 9 (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) → 𝑧𝐵)
82 imaeq2 5644 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑧) → (𝑓𝑦) = (𝑓 “ (𝑓𝑧)))
83 f1orel 6323 . . . . . . . . . . . . . . . 16 (𝑓:𝐴1-1-onto𝐵 → Rel 𝑓)
84 dfrel2 5766 . . . . . . . . . . . . . . . 16 (Rel 𝑓𝑓 = 𝑓)
8583, 84sylib 209 . . . . . . . . . . . . . . 15 (𝑓:𝐴1-1-onto𝐵𝑓 = 𝑓)
8685imaeq1d 5647 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1-onto𝐵 → (𝑓 “ (𝑓𝑧)) = (𝑓 “ (𝑓𝑧)))
8786adantr 472 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1-onto𝐵𝑧𝐵) → (𝑓 “ (𝑓𝑧)) = (𝑓 “ (𝑓𝑧)))
8846, 47syl 17 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1𝐴)
89 f1imacnv 6336 . . . . . . . . . . . . . 14 ((𝑓:𝐵1-1𝐴𝑧𝐵) → (𝑓 “ (𝑓𝑧)) = 𝑧)
9088, 89sylan 575 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1-onto𝐵𝑧𝐵) → (𝑓 “ (𝑓𝑧)) = 𝑧)
9187, 90eqtr3d 2801 . . . . . . . . . . . 12 ((𝑓:𝐴1-1-onto𝐵𝑧𝐵) → (𝑓 “ (𝑓𝑧)) = 𝑧)
9282, 91sylan9eqr 2821 . . . . . . . . . . 11 (((𝑓:𝐴1-1-onto𝐵𝑧𝐵) ∧ 𝑦 = (𝑓𝑧)) → (𝑓𝑦) = 𝑧)
9392eqcomd 2771 . . . . . . . . . 10 (((𝑓:𝐴1-1-onto𝐵𝑧𝐵) ∧ 𝑦 = (𝑓𝑧)) → 𝑧 = (𝑓𝑦))
9493ex 401 . . . . . . . . 9 ((𝑓:𝐴1-1-onto𝐵𝑧𝐵) → (𝑦 = (𝑓𝑧) → 𝑧 = (𝑓𝑦)))
9581, 94sylan2 586 . . . . . . . 8 ((𝑓:𝐴1-1-onto𝐵𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶})) → (𝑦 = (𝑓𝑧) → 𝑧 = (𝑓𝑦)))
9695adantrl 707 . . . . . . 7 ((𝑓:𝐴1-1-onto𝐵 ∧ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))) → (𝑦 = (𝑓𝑧) → 𝑧 = (𝑓𝑦)))
97 simpl 474 . . . . . . . . . . 11 ((𝑦 ∈ 𝒫 𝐴𝑦 ∈ {𝑥𝑥𝐶}) → 𝑦 ∈ 𝒫 𝐴)
9897elpwid 4327 . . . . . . . . . 10 ((𝑦 ∈ 𝒫 𝐴𝑦 ∈ {𝑥𝑥𝐶}) → 𝑦𝐴)
9932, 98sylbi 208 . . . . . . . . 9 (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) → 𝑦𝐴)
100 imaeq2 5644 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑦) → (𝑓𝑧) = (𝑓 “ (𝑓𝑦)))
101 f1imacnv 6336 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1𝐵𝑦𝐴) → (𝑓 “ (𝑓𝑦)) = 𝑦)
10217, 101sylan 575 . . . . . . . . . . . 12 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → (𝑓 “ (𝑓𝑦)) = 𝑦)
103100, 102sylan9eqr 2821 . . . . . . . . . . 11 (((𝑓:𝐴1-1-onto𝐵𝑦𝐴) ∧ 𝑧 = (𝑓𝑦)) → (𝑓𝑧) = 𝑦)
104103eqcomd 2771 . . . . . . . . . 10 (((𝑓:𝐴1-1-onto𝐵𝑦𝐴) ∧ 𝑧 = (𝑓𝑦)) → 𝑦 = (𝑓𝑧))
105104ex 401 . . . . . . . . 9 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → (𝑧 = (𝑓𝑦) → 𝑦 = (𝑓𝑧)))
10699, 105sylan2 586 . . . . . . . 8 ((𝑓:𝐴1-1-onto𝐵𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶})) → (𝑧 = (𝑓𝑦) → 𝑦 = (𝑓𝑧)))
107106adantrr 708 . . . . . . 7 ((𝑓:𝐴1-1-onto𝐵 ∧ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))) → (𝑧 = (𝑓𝑦) → 𝑦 = (𝑓𝑧)))
10896, 107impbid 203 . . . . . 6 ((𝑓:𝐴1-1-onto𝐵 ∧ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))) → (𝑦 = (𝑓𝑧) ↔ 𝑧 = (𝑓𝑦)))
109108ex 401 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → ((𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶})) → (𝑦 = (𝑓𝑧) ↔ 𝑧 = (𝑓𝑦))))
1108, 16, 45, 78, 109en3d 8197 . . . 4 (𝑓:𝐴1-1-onto𝐵 → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))
111110exlimiv 2025 . . 3 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))
1121, 111sylbi 208 . 2 (𝐴𝐵 → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))
113 df-pw 4317 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
114113ineq1i 3972 . . 3 (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) = ({𝑥𝑥𝐴} ∩ {𝑥𝑥𝐶})
115 inab 4059 . . 3 ({𝑥𝑥𝐴} ∩ {𝑥𝑥𝐶}) = {𝑥 ∣ (𝑥𝐴𝑥𝐶)}
116114, 115eqtri 2787 . 2 (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) = {𝑥 ∣ (𝑥𝐴𝑥𝐶)}
117 df-pw 4317 . . . 4 𝒫 𝐵 = {𝑥𝑥𝐵}
118117ineq1i 3972 . . 3 (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) = ({𝑥𝑥𝐵} ∩ {𝑥𝑥𝐶})
119 inab 4059 . . 3 ({𝑥𝑥𝐵} ∩ {𝑥𝑥𝐶}) = {𝑥 ∣ (𝑥𝐵𝑥𝐶)}
120118, 119eqtri 2787 . 2 (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) = {𝑥 ∣ (𝑥𝐵𝑥𝐶)}
121112, 116, 1203brtr3g 4842 1 (𝐴𝐵 → {𝑥 ∣ (𝑥𝐴𝑥𝐶)} ≈ {𝑥 ∣ (𝑥𝐵𝑥𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  {cab 2751  Vcvv 3350  cin 3731  wss 3732  𝒫 cpw 4315   class class class wbr 4809  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280  Rel wrel 5282  1-1wf1 6065  ontowfo 6066  1-1-ontowf1o 6067  cen 8157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-er 7947  df-en 8161
This theorem is referenced by:  infmap2  9293
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