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Theorem ofoafo 43374
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 4226 . . . . . 6 (𝐴𝐴) = 𝐴
32eqcomi 2745 . . . . 5 𝐴 = (𝐴𝐴)
43a1i 11 . . . 4 (𝐴𝑉𝐴 = (𝐴𝐴))
51, 1, 43jca 1128 . . 3 (𝐴𝑉 → (𝐴𝑉𝐴𝑉𝐴 = (𝐴𝐴)))
6 ofoaf 43373 . . 3 (((𝐴𝑉𝐴𝑉𝐴 = (𝐴𝐴)) ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))⟶(𝐶m 𝐴))
75, 6sylan 580 . 2 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴))):((𝐶m 𝐴) × (𝐶m 𝐴))⟶(𝐶m 𝐴))
8 simpr 484 . . . . 5 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ∈ (𝐶m 𝐴))
9 omelon 9687 . . . . . . . . . . . . . . 15 ω ∈ On
109a1i 11 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ω ∈ On)
11 simpl 482 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐵 ∈ On)
1210, 11jca 511 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ∈ On ∧ 𝐵 ∈ On))
13 peano1 7911 . . . . . . . . . . . . 13 ∅ ∈ ω
14 oen0 8625 . . . . . . . . . . . . 13 (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
1512, 13, 14sylancl 586 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ (ω ↑o 𝐵))
16 simpr 484 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 = (ω ↑o 𝐵))
1715, 16eleqtrrd 2843 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ 𝐶)
18 fconst6g 6796 . . . . . . . . . . 11 (∅ ∈ 𝐶 → (𝐴 × {∅}):𝐴𝐶)
1917, 18syl 17 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (𝐴 × {∅}):𝐴𝐶)
2019adantl 481 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}):𝐴𝐶)
21 oecl 8576 . . . . . . . . . . . . 13 ((ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑o 𝐵) ∈ On)
229, 11, 21sylancr 587 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ↑o 𝐵) ∈ On)
2316, 22eqeltrd 2840 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 ∈ On)
2423adantl 481 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐶 ∈ On)
25 simpl 482 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐴𝑉)
2624, 25elmapd 8881 . . . . . . . . 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 480 . . . . . . 7 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → (𝐴 × {∅}) ∈ (𝐶m 𝐴))
29 ovres 7600 . . . . . . . . . 10 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅})) = (f +o (𝐴 × {∅})))
3029adantl 481 . . . . . . . . 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 8890 . . . . . . . . . . . . . 14 ( ∈ (𝐶m 𝐴) → :𝐴𝐶)
3231adantr 480 . . . . . . . . . . . . 13 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → :𝐴𝐶)
3332ffnd 6736 . . . . . . . . . . . 12 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → Fn 𝐴)
3433adantl 481 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → Fn 𝐴)
35 elmapi 8890 . . . . . . . . . . . . . 14 ((𝐴 × {∅}) ∈ (𝐶m 𝐴) → (𝐴 × {∅}):𝐴𝐶)
3635adantl 481 . . . . . . . . . . . . 13 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → (𝐴 × {∅}):𝐴𝐶)
3736ffnd 6736 . . . . . . . . . . . 12 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
3837adantl 481 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → (𝐴 × {∅}) Fn 𝐴)
3925adantr 480 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → 𝐴𝑉)
4034, 38, 39, 39, 2offn 7711 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → (f +o (𝐴 × {∅})) Fn 𝐴)
41 elmapfn 8906 . . . . . . . . . . . . . 14 ( ∈ (𝐶m 𝐴) → Fn 𝐴)
42 elmapfn 8906 . . . . . . . . . . . . . 14 ((𝐴 × {∅}) ∈ (𝐶m 𝐴) → (𝐴 × {∅}) Fn 𝐴)
4341, 42anim12i 613 . . . . . . . . . . . . 13 (( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴)) → ( Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
4443adantl 481 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → ( Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
4539anim1i 615 . . . . . . . . . . . 12 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → (𝐴𝑉𝑎𝐴))
46 fnfvof 7715 . . . . . . . . . . . 12 ((( Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴) ∧ (𝐴𝑉𝑎𝐴)) → ((f +o (𝐴 × {∅}))‘𝑎) = ((𝑎) +o ((𝐴 × {∅})‘𝑎)))
4744, 45, 46syl2an2r 685 . . . . . . . . . . 11 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((f +o (𝐴 × {∅}))‘𝑎) = ((𝑎) +o ((𝐴 × {∅})‘𝑎)))
48 simpr 484 . . . . . . . . . . . . 13 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → 𝑎𝐴)
49 fvconst2g 7223 . . . . . . . . . . . . 13 ((∅ ∈ ω ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
5013, 48, 49sylancr 587 . . . . . . . . . . . 12 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
5150oveq2d 7448 . . . . . . . . . . 11 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((𝑎) +o ((𝐴 × {∅})‘𝑎)) = ((𝑎) +o ∅))
5224adantr 480 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → 𝐶 ∈ On)
5352adantr 480 . . . . . . . . . . . . . 14 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → 𝐶 ∈ On)
54 onss 7806 . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → :𝐴𝐶)
5756ffvelcdmda 7103 . . . . . . . . . . . . 13 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → (𝑎) ∈ 𝐶)
5855, 57sseldd 3983 . . . . . . . . . . . 12 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → (𝑎) ∈ On)
59 oa0 8555 . . . . . . . . . . . 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 2780 . . . . . . . . . 10 ((((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) ∧ 𝑎𝐴) → ((f +o (𝐴 × {∅}))‘𝑎) = (𝑎))
6240, 34, 61eqfnfvd 7053 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → (f +o (𝐴 × {∅})) = )
6330, 62eqtr2d 2777 . . . . . . . 8 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ( ∈ (𝐶m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶m 𝐴))) → = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅})))
6463expr 456 . . . . . . 7 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ((𝐴 × {∅}) ∈ (𝐶m 𝐴) → = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅}))))
6528, 64jcai 516 . . . . . 6 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ((𝐴 × {∅}) ∈ (𝐶m 𝐴) ∧ = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅}))))
66 oveq2 7440 . . . . . . 7 (𝑧 = (𝐴 × {∅}) → (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))(𝐴 × {∅})))
6766rspceeqv 3644 . . . . . 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 511 . . . 4 (((𝐴𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ∈ (𝐶m 𝐴)) → ( ∈ (𝐶m 𝐴) ∧ ∃𝑧 ∈ (𝐶m 𝐴) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)))
70 oveq1 7439 . . . . . . 7 (𝑓 = → (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧))
7170eqeq2d 2747 . . . . . 6 (𝑓 = → ( = (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) ↔ = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)))
7271rexbidv 3178 . . . . 5 (𝑓 = → (∃𝑧 ∈ (𝐶m 𝐴) = (𝑓( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧) ↔ ∃𝑧 ∈ (𝐶m 𝐴) = (( ∘f +o ↾ ((𝐶m 𝐴) × (𝐶m 𝐴)))𝑧)))
7372rspcev 3621 . . . 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 3145 . 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 7608 . 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 583 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 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  cin 3949  wss 3950  c0 4332  {csn 4625   × cxp 5682  cres 5686  Oncon0 6383   Fn wfn 6555  wf 6556  ontowfo 6558  cfv 6560  (class class class)co 7432  f cof 7696  ωcom 7888   +o coa 8504  o coe 8506  m cmap 8867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-oexp 8513  df-map 8869
This theorem is referenced by: (None)
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