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Theorem mndpsuppss 44788
 Description: The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
Hypothesis
Ref Expression
mndpsuppss.r 𝑅 = (Base‘𝑀)
Assertion
Ref Expression
mndpsuppss (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ⊆ ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))))

Proof of Theorem mndpsuppss
Dummy variables 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioran 981 . . . . . 6 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ (¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)))
2 nne 2991 . . . . . . 7 (¬ (𝐴𝑥) ≠ (0g𝑀) ↔ (𝐴𝑥) = (0g𝑀))
3 nne 2991 . . . . . . 7 (¬ (𝐵𝑥) ≠ (0g𝑀) ↔ (𝐵𝑥) = (0g𝑀))
42, 3anbi12i 629 . . . . . 6 ((¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
51, 4bitri 278 . . . . 5 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
6 elmapfn 8414 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
76ad2antrl 727 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐴 Fn 𝑉)
87adantr 484 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐴 Fn 𝑉)
9 elmapfn 8414 . . . . . . . . . . . 12 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
109ad2antll 728 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 Fn 𝑉)
1110adantr 484 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐵 Fn 𝑉)
12 simplr 768 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝑉𝑋)
1312adantr 484 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝑉𝑋)
14 inidm 4145 . . . . . . . . . 10 (𝑉𝑉) = 𝑉
15 simplrl 776 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐴𝑥) = (0g𝑀))
16 simplrr 777 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐵𝑥) = (0g𝑀))
178, 11, 13, 13, 14, 15, 16ofval 7399 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → ((𝐴f (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
1817an32s 651 . . . . . . . 8 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴f (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
19 eqid 2798 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
20 eqid 2798 . . . . . . . . . . . 12 (0g𝑀) = (0g𝑀)
2119, 20mndidcl 17920 . . . . . . . . . . 11 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
2221ancli 552 . . . . . . . . . 10 (𝑀 ∈ Mnd → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
2322ad4antr 731 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
24 eqid 2798 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
2519, 24, 20mndlid 17925 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)) → ((0g𝑀)(+g𝑀)(0g𝑀)) = (0g𝑀))
2623, 25syl 17 . . . . . . . 8 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((0g𝑀)(+g𝑀)(0g𝑀)) = (0g𝑀))
2718, 26eqtrd 2833 . . . . . . 7 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴f (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
28 nne 2991 . . . . . . 7 (¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀) ↔ ((𝐴f (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
2927, 28sylibr 237 . . . . . 6 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀))
3029ex 416 . . . . 5 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)) → ¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)))
315, 30syl5bi 245 . . . 4 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) → ¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)))
3231con4d 115 . . 3 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀) → ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))))
3332ss2rabdv 4003 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → {𝑥𝑉 ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))})
347, 10, 12, 12offun 7402 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → Fun (𝐴f (+g𝑀)𝐵))
35 ovexd 7170 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f (+g𝑀)𝐵) ∈ V)
36 fvexd 6660 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (0g𝑀) ∈ V)
37 suppval1 7821 . . . 4 ((Fun (𝐴f (+g𝑀)𝐵) ∧ (𝐴f (+g𝑀)𝐵) ∈ V ∧ (0g𝑀) ∈ V) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥 ∈ dom (𝐴f (+g𝑀)𝐵) ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
3834, 35, 36, 37syl3anc 1368 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥 ∈ dom (𝐴f (+g𝑀)𝐵) ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
3912, 7, 10offvalfv 44759 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f (+g𝑀)𝐵) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
4039dmeqd 5738 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → dom (𝐴f (+g𝑀)𝐵) = dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
41 ovex 7168 . . . . . 6 ((𝐴𝑣)(+g𝑀)(𝐵𝑣)) ∈ V
42 eqid 2798 . . . . . 6 (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣)))
4341, 42dmmpti 6464 . . . . 5 dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = 𝑉
4440, 43eqtrdi 2849 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → dom (𝐴f (+g𝑀)𝐵) = 𝑉)
4544rabeqdv 3432 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → {𝑥 ∈ dom (𝐴f (+g𝑀)𝐵) ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
4638, 45eqtrd 2833 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥𝑉 ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
47 elmapfun 8415 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → Fun 𝐴)
48 id 22 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 ∈ (𝑅m 𝑉))
49 fvexd 6660 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → (0g𝑀) ∈ V)
50 suppval1 7821 . . . . . . 7 ((Fun 𝐴𝐴 ∈ (𝑅m 𝑉) ∧ (0g𝑀) ∈ V) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5147, 48, 49, 50syl3anc 1368 . . . . . 6 (𝐴 ∈ (𝑅m 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
52 elmapi 8413 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → 𝐴:𝑉𝑅)
53 fdm 6495 . . . . . . 7 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
54 rabeq 3431 . . . . . . 7 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5552, 53, 543syl 18 . . . . . 6 (𝐴 ∈ (𝑅m 𝑉) → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5651, 55eqtrd 2833 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5756ad2antrl 727 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
58 elmapfun 8415 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → Fun 𝐵)
5958ad2antll 728 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → Fun 𝐵)
60 simprr 772 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 ∈ (𝑅m 𝑉))
61 suppval1 7821 . . . . . 6 ((Fun 𝐵𝐵 ∈ (𝑅m 𝑉) ∧ (0g𝑀) ∈ V) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6259, 60, 36, 61syl3anc 1368 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
63 elmapi 8413 . . . . . . . 8 (𝐵 ∈ (𝑅m 𝑉) → 𝐵:𝑉𝑅)
6463fdmd 6497 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → dom 𝐵 = 𝑉)
6564ad2antll 728 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → dom 𝐵 = 𝑉)
6665rabeqdv 3432 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6762, 66eqtrd 2833 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6857, 67uneq12d 4091 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}))
69 unrab 4226 . . 3 ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))}
7068, 69eqtrdi 2849 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))})
7133, 46, 703sstr4d 3962 1 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ⊆ ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881   ↦ cmpt 5110  dom cdm 5519  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘f cof 7388   supp csupp 7815   ↑m cmap 8391  Basecbs 16477  +gcplusg 16559  0gc0g 16707  Mndcmnd 17905 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7390  df-1st 7673  df-2nd 7674  df-supp 7816  df-map 8393  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906 This theorem is referenced by:  mndpsuppfi  44792
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