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Theorem ofoafo 42410
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 2739 . . . . 5 𝐴 = (𝐴𝐴)
43a1i 11 . . . 4 (𝐴𝑉𝐴 = (𝐴𝐴))
51, 1, 43jca 1126 . . 3 (𝐴𝑉 → (𝐴𝑉𝐴𝑉𝐴 = (𝐴𝐴)))
6 ofoaf 42409 . . 3 (((𝐴𝑉𝐴𝑉𝐴 = (𝐴𝐴)) ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))⟶(𝐶m 𝐴))
75, 6sylan 578 . 2 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))⟶(𝐶m 𝐴))
8 simpr 483 . . . . 5 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ∈ (𝐶m 𝐴))
9 omelon 9645 . . . . . . . . . . . . . . 15 ω ∈ On
109a1i 11 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ω ∈ On)
11 simpl 481 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐵 ∈ On)
1210, 11jca 510 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ∈ On ∧ 𝐵 ∈ On))
13 peano1 7883 . . . . . . . . . . . . 13 ∅ ∈ ω
14 oen0 8590 . . . . . . . . . . . . 13 (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
1512, 13, 14sylancl 584 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ (ω ↑o 𝐵))
16 simpr 483 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 = (ω ↑o 𝐵))
1715, 16eleqtrrd 2834 . . . . . . . . . . 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 480 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}):𝐴𝐶)
21 oecl 8541 . . . . . . . . . . . . 13 ((ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑o 𝐵) ∈ On)
229, 11, 21sylancr 585 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ↑o 𝐵) ∈ On)
2316, 22eqeltrd 2831 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 ∈ On)
2423adantl 480 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐶 ∈ On)
25 simpl 481 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐴𝑉)
2624, 25elmapd 8838 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ((𝐴 × {∅}) ∈ (𝐶m 𝐴) ↔ (𝐴 × {∅}):𝐴𝐶))
2720, 26mpbird 256 . . . . . . . 8 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}) ∈ (𝐶m 𝐴))
2827adantr 479 . . . . . . 7 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → (𝐴 × {∅}) ∈ (𝐶m 𝐴))
29 ovres 7577 . . . . . . . . . 10 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅})) = (f +o (𝐴 × {∅})))
3029adantl 480 . . . . . . . . 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 8847 . . . . . . . . . . . . . 14 ( ∈ (𝐶m 𝐴) → :𝐴𝐶)
3231adantr 479 . . . . . . . . . . . . 13 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → :𝐴𝐶)
3332ffnd 6719 . . . . . . . . . . . 12 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → Fn 𝐴)
3433adantl 480 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → Fn 𝐴)
35 elmapi 8847 . . . . . . . . . . . . . 14 ((𝐴 × {∅}) ∈ (𝐶m 𝐴) → (𝐴 × {∅}):𝐴𝐶)
3635adantl 480 . . . . . . . . . . . . 13 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → (𝐴 × {∅}):𝐴𝐶)
3736ffnd 6719 . . . . . . . . . . . 12 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
3837adantl 480 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → (𝐴 × {∅}) Fn 𝐴)
3925adantr 479 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → 𝐴𝑉)
4034, 38, 39, 39, 2offn 7687 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → (f +o (𝐴 × {∅})) Fn 𝐴)
41 elmapfn 8863 . . . . . . . . . . . . . 14 ( ∈ (𝐶m 𝐴) → Fn 𝐴)
42 elmapfn 8863 . . . . . . . . . . . . . 14 ((𝐴 × {∅}) ∈ (𝐶m 𝐴) → (𝐴 × {∅}) Fn 𝐴)
4341, 42anim12i 611 . . . . . . . . . . . . 13 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → ( Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
4443adantl 480 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → ( Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
4539anim1i 613 . . . . . . . . . . . 12 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → (𝐴𝑉𝑎𝐴))
46 fnfvof 7691 . . . . . . . . . . . 12 ((( Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴) ∧ (𝐴𝑉𝑎𝐴)) → ((f +o (𝐴 × {∅}))‘𝑎) = ((𝑎) +o ((𝐴 × {∅})‘𝑎)))
4744, 45, 46syl2an2r 681 . . . . . . . . . . 11 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((f +o (𝐴 × {∅}))‘𝑎) = ((𝑎) +o ((𝐴 × {∅})‘𝑎)))
48 simpr 483 . . . . . . . . . . . . 13 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → 𝑎𝐴)
49 fvconst2g 7206 . . . . . . . . . . . . 13 ((∅ ∈ ω ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
5013, 48, 49sylancr 585 . . . . . . . . . . . 12 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
5150oveq2d 7429 . . . . . . . . . . 11 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((𝑎) +o ((𝐴 × {∅})‘𝑎)) = ((𝑎) +o ∅))
5224adantr 479 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → 𝐶 ∈ On)
5352adantr 479 . . . . . . . . . . . . . 14 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → 𝐶 ∈ On)
54 onss 7776 . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 8520 . . . . . . . . . . . 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 2774 . . . . . . . . . 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 2771 . . . . . . . 8 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅})))
6463expr 455 . . . . . . 7 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ((𝐴 × {∅}) ∈ (𝐶m 𝐴) → = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅}))))
6528, 64jcai 515 . . . . . 6 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ((𝐴 × {∅}) ∈ (𝐶m 𝐴) ∧ = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅}))))
66 oveq2 7421 . . . . . . 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 510 . . . 4 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ( ∈ (𝐶m 𝐴) ∧ ∃𝑧 ∈ (𝐶m 𝐴) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)))
70 oveq1 7420 . . . . . . 7 (𝑓 = → (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧))
7170eqeq2d 2741 . . . . . 6 (𝑓 = → ( = (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) ↔ = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)))
7271rexbidv 3176 . . . . 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 3144 . 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 7585 . 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 581 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 394  w3a 1085   = wceq 1539  wcel 2104  wral 3059  wrex 3068  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 7413  f cof 7672  ωcom 7859   +o coa 8467  o coe 8469  m cmap 8824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-inf2 9640
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 7416  df-oprab 7417  df-mpo 7418  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-omul 8475  df-oexp 8476  df-map 8826
This theorem is referenced by: (None)
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