MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssenen Structured version   Visualization version   GIF version

Theorem ssenen 8175
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 8006 . . 3 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
2 f1odm 6179 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → dom 𝑓 = 𝐴)
3 vex 3234 . . . . . . . 8 𝑓 ∈ V
43dmex 7141 . . . . . . 7 dom 𝑓 ∈ V
52, 4syl6eqelr 2739 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝐴 ∈ V)
6 pwexg 4880 . . . . . 6 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
7 inex1g 4834 . . . . . 6 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∈ V)
85, 6, 73syl 18 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∈ V)
9 f1ofo 6182 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
10 forn 6156 . . . . . . . 8 (𝑓:𝐴onto𝐵 → ran 𝑓 = 𝐵)
119, 10syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → ran 𝑓 = 𝐵)
123rnex 7142 . . . . . . 7 ran 𝑓 ∈ V
1311, 12syl6eqelr 2739 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝐵 ∈ V)
14 pwexg 4880 . . . . . 6 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
15 inex1g 4834 . . . . . 6 (𝒫 𝐵 ∈ V → (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ∈ V)
1613, 14, 153syl 18 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ∈ V)
17 f1of1 6174 . . . . . . . . . . 11 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴1-1𝐵)
1817adantr 480 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → 𝑓:𝐴1-1𝐵)
1913adantr 480 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → 𝐵 ∈ V)
20 simpr 476 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → 𝑦𝐴)
21 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
2221a1i 11 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → 𝑦 ∈ V)
23 f1imaen2g 8058 . . . . . . . . . 10 (((𝑓:𝐴1-1𝐵𝐵 ∈ V) ∧ (𝑦𝐴𝑦 ∈ V)) → (𝑓𝑦) ≈ 𝑦)
2418, 19, 20, 22, 23syl22anc 1367 . . . . . . . . 9 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → (𝑓𝑦) ≈ 𝑦)
25 entr 8049 . . . . . . . . 9 (((𝑓𝑦) ≈ 𝑦𝑦𝐶) → (𝑓𝑦) ≈ 𝐶)
2624, 25sylan 487 . . . . . . . 8 (((𝑓:𝐴1-1-onto𝐵𝑦𝐴) ∧ 𝑦𝐶) → (𝑓𝑦) ≈ 𝐶)
2726expl 647 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → ((𝑦𝐴𝑦𝐶) → (𝑓𝑦) ≈ 𝐶))
28 imassrn 5512 . . . . . . . . 9 (𝑓𝑦) ⊆ ran 𝑓
2928, 10syl5sseq 3686 . . . . . . . 8 (𝑓:𝐴onto𝐵 → (𝑓𝑦) ⊆ 𝐵)
309, 29syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓𝑦) ⊆ 𝐵)
3127, 30jctild 565 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ((𝑦𝐴𝑦𝐶) → ((𝑓𝑦) ⊆ 𝐵 ∧ (𝑓𝑦) ≈ 𝐶)))
32 elin 3829 . . . . . . 7 (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ↔ (𝑦 ∈ 𝒫 𝐴𝑦 ∈ {𝑥𝑥𝐶}))
3321elpw 4197 . . . . . . . 8 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
34 breq1 4688 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐶𝑦𝐶))
3521, 34elab 3382 . . . . . . . 8 (𝑦 ∈ {𝑥𝑥𝐶} ↔ 𝑦𝐶)
3633, 35anbi12i 733 . . . . . . 7 ((𝑦 ∈ 𝒫 𝐴𝑦 ∈ {𝑥𝑥𝐶}) ↔ (𝑦𝐴𝑦𝐶))
3732, 36bitri 264 . . . . . 6 (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ↔ (𝑦𝐴𝑦𝐶))
38 elin 3829 . . . . . . 7 ((𝑓𝑦) ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ↔ ((𝑓𝑦) ∈ 𝒫 𝐵 ∧ (𝑓𝑦) ∈ {𝑥𝑥𝐶}))
393imaex 7146 . . . . . . . . 9 (𝑓𝑦) ∈ V
4039elpw 4197 . . . . . . . 8 ((𝑓𝑦) ∈ 𝒫 𝐵 ↔ (𝑓𝑦) ⊆ 𝐵)
41 breq1 4688 . . . . . . . . 9 (𝑥 = (𝑓𝑦) → (𝑥𝐶 ↔ (𝑓𝑦) ≈ 𝐶))
4239, 41elab 3382 . . . . . . . 8 ((𝑓𝑦) ∈ {𝑥𝑥𝐶} ↔ (𝑓𝑦) ≈ 𝐶)
4340, 42anbi12i 733 . . . . . . 7 (((𝑓𝑦) ∈ 𝒫 𝐵 ∧ (𝑓𝑦) ∈ {𝑥𝑥𝐶}) ↔ ((𝑓𝑦) ⊆ 𝐵 ∧ (𝑓𝑦) ≈ 𝐶))
4438, 43bitri 264 . . . . . 6 ((𝑓𝑦) ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ↔ ((𝑓𝑦) ⊆ 𝐵 ∧ (𝑓𝑦) ≈ 𝐶))
4531, 37, 443imtr4g 285 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) → (𝑓𝑦) ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶})))
46 f1ocnv 6187 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
47 f1of1 6174 . . . . . . . . . . . 12 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵1-1𝐴)
48 f1f1orn 6186 . . . . . . . . . . . 12 (𝑓:𝐵1-1𝐴𝑓:𝐵1-1-onto→ran 𝑓)
49 f1of1 6174 . . . . . . . . . . . 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 3234 . . . . . . . . . . . 12 𝑧 ∈ V
5251f1imaen 8059 . . . . . . . . . . 11 ((𝑓:𝐵1-1→ran 𝑓𝑧𝐵) → (𝑓𝑧) ≈ 𝑧)
5350, 52sylan 487 . . . . . . . . . 10 ((𝑓:𝐵1-1-onto𝐴𝑧𝐵) → (𝑓𝑧) ≈ 𝑧)
54 entr 8049 . . . . . . . . . 10 (((𝑓𝑧) ≈ 𝑧𝑧𝐶) → (𝑓𝑧) ≈ 𝐶)
5553, 54sylan 487 . . . . . . . . 9 (((𝑓:𝐵1-1-onto𝐴𝑧𝐵) ∧ 𝑧𝐶) → (𝑓𝑧) ≈ 𝐶)
5655expl 647 . . . . . . . 8 (𝑓:𝐵1-1-onto𝐴 → ((𝑧𝐵𝑧𝐶) → (𝑓𝑧) ≈ 𝐶))
57 f1ofo 6182 . . . . . . . . 9 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵onto𝐴)
58 imassrn 5512 . . . . . . . . . 10 (𝑓𝑧) ⊆ ran 𝑓
59 forn 6156 . . . . . . . . . 10 (𝑓:𝐵onto𝐴 → ran 𝑓 = 𝐴)
6058, 59syl5sseq 3686 . . . . . . . . 9 (𝑓:𝐵onto𝐴 → (𝑓𝑧) ⊆ 𝐴)
6157, 60syl 17 . . . . . . . 8 (𝑓:𝐵1-1-onto𝐴 → (𝑓𝑧) ⊆ 𝐴)
6256, 61jctild 565 . . . . . . 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 3829 . . . . . . 7 (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ↔ (𝑧 ∈ 𝒫 𝐵𝑧 ∈ {𝑥𝑥𝐶}))
6551elpw 4197 . . . . . . . 8 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
66 breq1 4688 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝐶𝑧𝐶))
6751, 66elab 3382 . . . . . . . 8 (𝑧 ∈ {𝑥𝑥𝐶} ↔ 𝑧𝐶)
6865, 67anbi12i 733 . . . . . . 7 ((𝑧 ∈ 𝒫 𝐵𝑧 ∈ {𝑥𝑥𝐶}) ↔ (𝑧𝐵𝑧𝐶))
6964, 68bitri 264 . . . . . 6 (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) ↔ (𝑧𝐵𝑧𝐶))
70 elin 3829 . . . . . . 7 ((𝑓𝑧) ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ↔ ((𝑓𝑧) ∈ 𝒫 𝐴 ∧ (𝑓𝑧) ∈ {𝑥𝑥𝐶}))
713cnvex 7155 . . . . . . . . . 10 𝑓 ∈ V
7271imaex 7146 . . . . . . . . 9 (𝑓𝑧) ∈ V
7372elpw 4197 . . . . . . . 8 ((𝑓𝑧) ∈ 𝒫 𝐴 ↔ (𝑓𝑧) ⊆ 𝐴)
74 breq1 4688 . . . . . . . . 9 (𝑥 = (𝑓𝑧) → (𝑥𝐶 ↔ (𝑓𝑧) ≈ 𝐶))
7572, 74elab 3382 . . . . . . . 8 ((𝑓𝑧) ∈ {𝑥𝑥𝐶} ↔ (𝑓𝑧) ≈ 𝐶)
7673, 75anbi12i 733 . . . . . . 7 (((𝑓𝑧) ∈ 𝒫 𝐴 ∧ (𝑓𝑧) ∈ {𝑥𝑥𝐶}) ↔ ((𝑓𝑧) ⊆ 𝐴 ∧ (𝑓𝑧) ≈ 𝐶))
7770, 76bitri 264 . . . . . 6 ((𝑓𝑧) ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ↔ ((𝑓𝑧) ⊆ 𝐴 ∧ (𝑓𝑧) ≈ 𝐶))
7863, 69, 773imtr4g 285 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) → (𝑓𝑧) ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶})))
79 simpl 472 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 𝐵𝑧 ∈ {𝑥𝑥𝐶}) → 𝑧 ∈ 𝒫 𝐵)
8079elpwid 4203 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 𝐵𝑧 ∈ {𝑥𝑥𝐶}) → 𝑧𝐵)
8164, 80sylbi 207 . . . . . . . . 9 (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) → 𝑧𝐵)
82 imaeq2 5497 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑧) → (𝑓𝑦) = (𝑓 “ (𝑓𝑧)))
83 f1orel 6178 . . . . . . . . . . . . . . . 16 (𝑓:𝐴1-1-onto𝐵 → Rel 𝑓)
84 dfrel2 5618 . . . . . . . . . . . . . . . 16 (Rel 𝑓𝑓 = 𝑓)
8583, 84sylib 208 . . . . . . . . . . . . . . 15 (𝑓:𝐴1-1-onto𝐵𝑓 = 𝑓)
8685imaeq1d 5500 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1-onto𝐵 → (𝑓 “ (𝑓𝑧)) = (𝑓 “ (𝑓𝑧)))
8786adantr 480 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1-onto𝐵𝑧𝐵) → (𝑓 “ (𝑓𝑧)) = (𝑓 “ (𝑓𝑧)))
8846, 47syl 17 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1𝐴)
89 f1imacnv 6191 . . . . . . . . . . . . . 14 ((𝑓:𝐵1-1𝐴𝑧𝐵) → (𝑓 “ (𝑓𝑧)) = 𝑧)
9088, 89sylan 487 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1-onto𝐵𝑧𝐵) → (𝑓 “ (𝑓𝑧)) = 𝑧)
9187, 90eqtr3d 2687 . . . . . . . . . . . 12 ((𝑓:𝐴1-1-onto𝐵𝑧𝐵) → (𝑓 “ (𝑓𝑧)) = 𝑧)
9282, 91sylan9eqr 2707 . . . . . . . . . . 11 (((𝑓:𝐴1-1-onto𝐵𝑧𝐵) ∧ 𝑦 = (𝑓𝑧)) → (𝑓𝑦) = 𝑧)
9392eqcomd 2657 . . . . . . . . . 10 (((𝑓:𝐴1-1-onto𝐵𝑧𝐵) ∧ 𝑦 = (𝑓𝑧)) → 𝑧 = (𝑓𝑦))
9493ex 449 . . . . . . . . 9 ((𝑓:𝐴1-1-onto𝐵𝑧𝐵) → (𝑦 = (𝑓𝑧) → 𝑧 = (𝑓𝑦)))
9581, 94sylan2 490 . . . . . . . 8 ((𝑓:𝐴1-1-onto𝐵𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶})) → (𝑦 = (𝑓𝑧) → 𝑧 = (𝑓𝑦)))
9695adantrl 752 . . . . . . 7 ((𝑓:𝐴1-1-onto𝐵 ∧ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))) → (𝑦 = (𝑓𝑧) → 𝑧 = (𝑓𝑦)))
97 simpl 472 . . . . . . . . . . 11 ((𝑦 ∈ 𝒫 𝐴𝑦 ∈ {𝑥𝑥𝐶}) → 𝑦 ∈ 𝒫 𝐴)
9897elpwid 4203 . . . . . . . . . 10 ((𝑦 ∈ 𝒫 𝐴𝑦 ∈ {𝑥𝑥𝐶}) → 𝑦𝐴)
9932, 98sylbi 207 . . . . . . . . 9 (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) → 𝑦𝐴)
100 imaeq2 5497 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑦) → (𝑓𝑧) = (𝑓 “ (𝑓𝑦)))
101 f1imacnv 6191 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1𝐵𝑦𝐴) → (𝑓 “ (𝑓𝑦)) = 𝑦)
10217, 101sylan 487 . . . . . . . . . . . 12 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → (𝑓 “ (𝑓𝑦)) = 𝑦)
103100, 102sylan9eqr 2707 . . . . . . . . . . 11 (((𝑓:𝐴1-1-onto𝐵𝑦𝐴) ∧ 𝑧 = (𝑓𝑦)) → (𝑓𝑧) = 𝑦)
104103eqcomd 2657 . . . . . . . . . 10 (((𝑓:𝐴1-1-onto𝐵𝑦𝐴) ∧ 𝑧 = (𝑓𝑦)) → 𝑦 = (𝑓𝑧))
105104ex 449 . . . . . . . . 9 ((𝑓:𝐴1-1-onto𝐵𝑦𝐴) → (𝑧 = (𝑓𝑦) → 𝑦 = (𝑓𝑧)))
10699, 105sylan2 490 . . . . . . . 8 ((𝑓:𝐴1-1-onto𝐵𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶})) → (𝑧 = (𝑓𝑦) → 𝑦 = (𝑓𝑧)))
107106adantrr 753 . . . . . . 7 ((𝑓:𝐴1-1-onto𝐵 ∧ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))) → (𝑧 = (𝑓𝑦) → 𝑦 = (𝑓𝑧)))
10896, 107impbid 202 . . . . . 6 ((𝑓:𝐴1-1-onto𝐵 ∧ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))) → (𝑦 = (𝑓𝑧) ↔ 𝑧 = (𝑓𝑦)))
109108ex 449 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → ((𝑦 ∈ (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶})) → (𝑦 = (𝑓𝑧) ↔ 𝑧 = (𝑓𝑦))))
1108, 16, 45, 78, 109en3d 8034 . . . 4 (𝑓:𝐴1-1-onto𝐵 → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))
111110exlimiv 1898 . . 3 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))
1121, 111sylbi 207 . 2 (𝐴𝐵 → (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥𝑥𝐶}))
113 df-pw 4193 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
114113ineq1i 3843 . . 3 (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) = ({𝑥𝑥𝐴} ∩ {𝑥𝑥𝐶})
115 inab 3928 . . 3 ({𝑥𝑥𝐴} ∩ {𝑥𝑥𝐶}) = {𝑥 ∣ (𝑥𝐴𝑥𝐶)}
116114, 115eqtri 2673 . 2 (𝒫 𝐴 ∩ {𝑥𝑥𝐶}) = {𝑥 ∣ (𝑥𝐴𝑥𝐶)}
117 df-pw 4193 . . . 4 𝒫 𝐵 = {𝑥𝑥𝐵}
118117ineq1i 3843 . . 3 (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) = ({𝑥𝑥𝐵} ∩ {𝑥𝑥𝐶})
119 inab 3928 . . 3 ({𝑥𝑥𝐵} ∩ {𝑥𝑥𝐶}) = {𝑥 ∣ (𝑥𝐵𝑥𝐶)}
120118, 119eqtri 2673 . 2 (𝒫 𝐵 ∩ {𝑥𝑥𝐶}) = {𝑥 ∣ (𝑥𝐵𝑥𝐶)}
121112, 116, 1203brtr3g 4718 1 (𝐴𝐵 → {𝑥 ∣ (𝑥𝐴𝑥𝐶)} ≈ {𝑥 ∣ (𝑥𝐵𝑥𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191   class class class wbr 4685  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Rel wrel 5148  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cen 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998
This theorem is referenced by:  infmap2  9078
  Copyright terms: Public domain W3C validator