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Theorem ofoafo 42106
Description: Addition operator for functions from a set into a power of omega is an onto binary operator. (Contributed by RP, 5-Jan-2025.)
Assertion
Ref Expression
ofoafo ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))–onto→(𝐶m 𝐴))

Proof of Theorem ofoafo
Dummy variables 𝑎 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
2 inidm 4219 . . . . . 6 (𝐴𝐴) = 𝐴
32eqcomi 2742 . . . . 5 𝐴 = (𝐴𝐴)
43a1i 11 . . . 4 (𝐴𝑉𝐴 = (𝐴𝐴))
51, 1, 43jca 1129 . . 3 (𝐴𝑉 → (𝐴𝑉𝐴𝑉𝐴 = (𝐴𝐴)))
6 ofoaf 42105 . . 3 (((𝐴𝑉𝐴𝑉𝐴 = (𝐴𝐴)) ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))⟶(𝐶m 𝐴))
75, 6sylan 581 . 2 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))⟶(𝐶m 𝐴))
8 simpr 486 . . . . 5 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ∈ (𝐶m 𝐴))
9 omelon 9641 . . . . . . . . . . . . . . 15 ω ∈ On
109a1i 11 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ω ∈ On)
11 simpl 484 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐵 ∈ On)
1210, 11jca 513 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ∈ On ∧ 𝐵 ∈ On))
13 peano1 7879 . . . . . . . . . . . . 13 ∅ ∈ ω
14 oen0 8586 . . . . . . . . . . . . 13 (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
1512, 13, 14sylancl 587 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ (ω ↑o 𝐵))
16 simpr 486 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 = (ω ↑o 𝐵))
1715, 16eleqtrrd 2837 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ 𝐶)
18 fconst6g 6781 . . . . . . . . . . 11 (∅ ∈ 𝐶 → (𝐴 × {∅}):𝐴𝐶)
1917, 18syl 17 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (𝐴 × {∅}):𝐴𝐶)
2019adantl 483 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}):𝐴𝐶)
21 oecl 8537 . . . . . . . . . . . . 13 ((ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑o 𝐵) ∈ On)
229, 11, 21sylancr 588 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ↑o 𝐵) ∈ On)
2316, 22eqeltrd 2834 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 ∈ On)
2423adantl 483 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐶 ∈ On)
25 simpl 484 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐴𝑉)
2624, 25elmapd 8834 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ((𝐴 × {∅}) ∈ (𝐶m 𝐴) ↔ (𝐴 × {∅}):𝐴𝐶))
2720, 26mpbird 257 . . . . . . . 8 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}) ∈ (𝐶m 𝐴))
2827adantr 482 . . . . . . 7 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → (𝐴 × {∅}) ∈ (𝐶m 𝐴))
29 ovres 7573 . . . . . . . . . 10 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅})) = (f +o (𝐴 × {∅})))
3029adantl 483 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅})) = (f +o (𝐴 × {∅})))
31 elmapi 8843 . . . . . . . . . . . . . 14 ( ∈ (𝐶m 𝐴) → :𝐴𝐶)
3231adantr 482 . . . . . . . . . . . . 13 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → :𝐴𝐶)
3332ffnd 6719 . . . . . . . . . . . 12 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → Fn 𝐴)
3433adantl 483 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → Fn 𝐴)
35 elmapi 8843 . . . . . . . . . . . . . 14 ((𝐴 × {∅}) ∈ (𝐶m 𝐴) → (𝐴 × {∅}):𝐴𝐶)
3635adantl 483 . . . . . . . . . . . . 13 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → (𝐴 × {∅}):𝐴𝐶)
3736ffnd 6719 . . . . . . . . . . . 12 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
3837adantl 483 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → (𝐴 × {∅}) Fn 𝐴)
3925adantr 482 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → 𝐴𝑉)
4034, 38, 39, 39, 2offn 7683 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → (f +o (𝐴 × {∅})) Fn 𝐴)
41 elmapfn 8859 . . . . . . . . . . . . . 14 ( ∈ (𝐶m 𝐴) → Fn 𝐴)
42 elmapfn 8859 . . . . . . . . . . . . . 14 ((𝐴 × {∅}) ∈ (𝐶m 𝐴) → (𝐴 × {∅}) Fn 𝐴)
4341, 42anim12i 614 . . . . . . . . . . . . 13 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → ( Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
4443adantl 483 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → ( Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
4539anim1i 616 . . . . . . . . . . . 12 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → (𝐴𝑉𝑎𝐴))
46 fnfvof 7687 . . . . . . . . . . . 12 ((( Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴) ∧ (𝐴𝑉𝑎𝐴)) → ((f +o (𝐴 × {∅}))‘𝑎) = ((𝑎) +o ((𝐴 × {∅})‘𝑎)))
4744, 45, 46syl2an2r 684 . . . . . . . . . . 11 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((f +o (𝐴 × {∅}))‘𝑎) = ((𝑎) +o ((𝐴 × {∅})‘𝑎)))
48 simpr 486 . . . . . . . . . . . . 13 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → 𝑎𝐴)
49 fvconst2g 7203 . . . . . . . . . . . . 13 ((∅ ∈ ω ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
5013, 48, 49sylancr 588 . . . . . . . . . . . 12 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
5150oveq2d 7425 . . . . . . . . . . 11 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((𝑎) +o ((𝐴 × {∅})‘𝑎)) = ((𝑎) +o ∅))
5224adantr 482 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → 𝐶 ∈ On)
5352adantr 482 . . . . . . . . . . . . . 14 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → 𝐶 ∈ On)
54 onss 7772 . . . . . . . . . . . . . 14 (𝐶 ∈ On → 𝐶 ⊆ On)
5553, 54syl 17 . . . . . . . . . . . . 13 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → 𝐶 ⊆ On)
5631ad2antrl 727 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → :𝐴𝐶)
5756ffvelcdmda 7087 . . . . . . . . . . . . 13 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → (𝑎) ∈ 𝐶)
5855, 57sseldd 3984 . . . . . . . . . . . 12 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → (𝑎) ∈ On)
59 oa0 8516 . . . . . . . . . . . 12 ((𝑎) ∈ On → ((𝑎) +o ∅) = (𝑎))
6058, 59syl 17 . . . . . . . . . . 11 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((𝑎) +o ∅) = (𝑎))
6147, 51, 603eqtrd 2777 . . . . . . . . . 10 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((f +o (𝐴 × {∅}))‘𝑎) = (𝑎))
6240, 34, 61eqfnfvd 7036 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → (f +o (𝐴 × {∅})) = )
6330, 62eqtr2d 2774 . . . . . . . 8 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅})))
6463expr 458 . . . . . . 7 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ((𝐴 × {∅}) ∈ (𝐶m 𝐴) → = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅}))))
6528, 64jcai 518 . . . . . 6 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ((𝐴 × {∅}) ∈ (𝐶m 𝐴) ∧ = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅}))))
66 oveq2 7417 . . . . . . 7 (𝑧 = (𝐴 × {∅}) → (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅})))
6766rspceeqv 3634 . . . . . 6 (((𝐴 × {∅}) ∈ (𝐶m 𝐴) ∧ = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅}))) → ∃𝑧 ∈ (𝐶m 𝐴) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧))
6865, 67syl 17 . . . . 5 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ∃𝑧 ∈ (𝐶m 𝐴) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧))
698, 68jca 513 . . . 4 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ( ∈ (𝐶m 𝐴) ∧ ∃𝑧 ∈ (𝐶m 𝐴) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)))
70 oveq1 7416 . . . . . . 7 (𝑓 = → (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧))
7170eqeq2d 2744 . . . . . 6 (𝑓 = → ( = (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) ↔ = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)))
7271rexbidv 3179 . . . . 5 (𝑓 = → (∃𝑧 ∈ (𝐶m 𝐴) = (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) ↔ ∃𝑧 ∈ (𝐶m 𝐴) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)))
7372rspcev 3613 . . . 4 (( ∈ (𝐶m 𝐴) ∧ ∃𝑧 ∈ (𝐶m 𝐴) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)) → ∃𝑓 ∈ (𝐶m 𝐴)∃𝑧 ∈ (𝐶m 𝐴) = (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧))
7469, 73syl 17 . . 3 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ∃𝑓 ∈ (𝐶m 𝐴)∃𝑧 ∈ (𝐶m 𝐴) = (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧))
7574ralrimiva 3147 . 2 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ∀ ∈ (𝐶m 𝐴)∃𝑓 ∈ (𝐶m 𝐴)∃𝑧 ∈ (𝐶m 𝐴) = (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧))
76 foov 7581 . 2 (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))–onto→(𝐶m 𝐴) ↔ (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))⟶(𝐶m 𝐴) ∧ ∀ ∈ (𝐶m 𝐴)∃𝑓 ∈ (𝐶m 𝐴)∃𝑧 ∈ (𝐶m 𝐴) = (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)))
777, 75, 76sylanbrc 584 1 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))–onto→(𝐶m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cin 3948  wss 3949  c0 4323  {csn 4629   × cxp 5675  cres 5679  Oncon0 6365   Fn wfn 6539  wf 6540  ontowfo 6542  cfv 6544  (class class class)co 7409  f cof 7668  ωcom 7855   +o coa 8463  o coe 8465  m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-omul 8471  df-oexp 8472  df-map 8822
This theorem is referenced by: (None)
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