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Theorem fsovcnvlem 44002
Description: The 𝑂 operator, which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)
Hypotheses
Ref Expression
fsovd.fs 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
fsovd.a (𝜑𝐴𝑉)
fsovd.b (𝜑𝐵𝑊)
fsovfvd.g 𝐺 = (𝐴𝑂𝐵)
fsovcnvlem.h 𝐻 = (𝐵𝑂𝐴)
Assertion
Ref Expression
fsovcnvlem (𝜑 → (𝐻𝐺) = ( I ↾ (𝒫 𝐵m 𝐴)))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑓,𝑥,𝑦   𝐵,𝑎,𝑏,𝑓,𝑦   𝜑,𝑎,𝑏,𝑓,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐺(𝑥,𝑦,𝑓,𝑎,𝑏)   𝐻(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑓,𝑎,𝑏)

Proof of Theorem fsovcnvlem
Dummy variables 𝑐 𝑑 𝑔 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsovd.a . . . . . . . 8 (𝜑𝐴𝑉)
2 ssrab2 4089 . . . . . . . . 9 {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ⊆ 𝐴
32a1i 11 . . . . . . . 8 (𝜑 → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ⊆ 𝐴)
41, 3sselpwd 5333 . . . . . . 7 (𝜑 → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ∈ 𝒫 𝐴)
54adantr 480 . . . . . 6 ((𝜑𝑦𝐵) → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ∈ 𝒫 𝐴)
65fmpttd 7134 . . . . 5 (𝜑 → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}):𝐵⟶𝒫 𝐴)
71pwexd 5384 . . . . . 6 (𝜑 → 𝒫 𝐴 ∈ V)
8 fsovd.b . . . . . 6 (𝜑𝐵𝑊)
97, 8elmapd 8878 . . . . 5 (𝜑 → ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) ∈ (𝒫 𝐴m 𝐵) ↔ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}):𝐵⟶𝒫 𝐴))
106, 9mpbird 257 . . . 4 (𝜑 → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) ∈ (𝒫 𝐴m 𝐵))
1110adantr 480 . . 3 ((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) ∈ (𝒫 𝐴m 𝐵))
12 fsovfvd.g . . . 4 𝐺 = (𝐴𝑂𝐵)
13 fsovd.fs . . . . 5 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
1413, 1, 8fsovd 43997 . . . 4 (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
1512, 14eqtrid 2786 . . 3 (𝜑𝐺 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
16 fsovcnvlem.h . . . 4 𝐻 = (𝐵𝑂𝐴)
17 oveq2 7438 . . . . . . . 8 (𝑎 = 𝑑 → (𝒫 𝑏m 𝑎) = (𝒫 𝑏m 𝑑))
18 rabeq 3447 . . . . . . . . 9 (𝑎 = 𝑑 → {𝑥𝑎𝑦 ∈ (𝑓𝑥)} = {𝑥𝑑𝑦 ∈ (𝑓𝑥)})
1918mpteq2dv 5249 . . . . . . . 8 (𝑎 = 𝑑 → (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)}) = (𝑦𝑏 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)}))
2017, 19mpteq12dv 5238 . . . . . . 7 (𝑎 = 𝑑 → (𝑓 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})) = (𝑓 ∈ (𝒫 𝑏m 𝑑) ↦ (𝑦𝑏 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)})))
21 pweq 4618 . . . . . . . . 9 (𝑏 = 𝑐 → 𝒫 𝑏 = 𝒫 𝑐)
2221oveq1d 7445 . . . . . . . 8 (𝑏 = 𝑐 → (𝒫 𝑏m 𝑑) = (𝒫 𝑐m 𝑑))
23 mpteq1 5240 . . . . . . . 8 (𝑏 = 𝑐 → (𝑦𝑏 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)}) = (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)}))
2422, 23mpteq12dv 5238 . . . . . . 7 (𝑏 = 𝑐 → (𝑓 ∈ (𝒫 𝑏m 𝑑) ↦ (𝑦𝑏 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)})) = (𝑓 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)})))
2520, 24cbvmpov 7527 . . . . . 6 (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)}))) = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑓 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)})))
26 eqid 2734 . . . . . . 7 V = V
27 fveq1 6905 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓𝑥) = (𝑔𝑥))
2827eleq2d 2824 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑦 ∈ (𝑓𝑥) ↔ 𝑦 ∈ (𝑔𝑥)))
2928rabbidv 3440 . . . . . . . . . 10 (𝑓 = 𝑔 → {𝑥𝑑𝑦 ∈ (𝑓𝑥)} = {𝑥𝑑𝑦 ∈ (𝑔𝑥)})
3029mpteq2dv 5249 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)}) = (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑔𝑥)}))
3130cbvmptv 5260 . . . . . . . 8 (𝑓 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)})) = (𝑔 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑔𝑥)}))
32 eleq1w 2821 . . . . . . . . . . . 12 (𝑦 = 𝑢 → (𝑦 ∈ (𝑔𝑥) ↔ 𝑢 ∈ (𝑔𝑥)))
3332rabbidv 3440 . . . . . . . . . . 11 (𝑦 = 𝑢 → {𝑥𝑑𝑦 ∈ (𝑔𝑥)} = {𝑥𝑑𝑢 ∈ (𝑔𝑥)})
3433cbvmptv 5260 . . . . . . . . . 10 (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑔𝑥)}) = (𝑢𝑐 ↦ {𝑥𝑑𝑢 ∈ (𝑔𝑥)})
35 fveq2 6906 . . . . . . . . . . . . 13 (𝑥 = 𝑣 → (𝑔𝑥) = (𝑔𝑣))
3635eleq2d 2824 . . . . . . . . . . . 12 (𝑥 = 𝑣 → (𝑢 ∈ (𝑔𝑥) ↔ 𝑢 ∈ (𝑔𝑣)))
3736cbvrabv 3443 . . . . . . . . . . 11 {𝑥𝑑𝑢 ∈ (𝑔𝑥)} = {𝑣𝑑𝑢 ∈ (𝑔𝑣)}
3837mpteq2i 5252 . . . . . . . . . 10 (𝑢𝑐 ↦ {𝑥𝑑𝑢 ∈ (𝑔𝑥)}) = (𝑢𝑐 ↦ {𝑣𝑑𝑢 ∈ (𝑔𝑣)})
3934, 38eqtri 2762 . . . . . . . . 9 (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑔𝑥)}) = (𝑢𝑐 ↦ {𝑣𝑑𝑢 ∈ (𝑔𝑣)})
4039mpteq2i 5252 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑔𝑥)})) = (𝑔 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑢𝑐 ↦ {𝑣𝑑𝑢 ∈ (𝑔𝑣)}))
4131, 40eqtri 2762 . . . . . . 7 (𝑓 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)})) = (𝑔 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑢𝑐 ↦ {𝑣𝑑𝑢 ∈ (𝑔𝑣)}))
4226, 26, 41mpoeq123i 7508 . . . . . 6 (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑓 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑦𝑐 ↦ {𝑥𝑑𝑦 ∈ (𝑓𝑥)}))) = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑔 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑢𝑐 ↦ {𝑣𝑑𝑢 ∈ (𝑔𝑣)})))
4313, 25, 423eqtri 2766 . . . . 5 𝑂 = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑔 ∈ (𝒫 𝑐m 𝑑) ↦ (𝑢𝑐 ↦ {𝑣𝑑𝑢 ∈ (𝑔𝑣)})))
4443, 8, 1fsovd 43997 . . . 4 (𝜑 → (𝐵𝑂𝐴) = (𝑔 ∈ (𝒫 𝐴m 𝐵) ↦ (𝑢𝐴 ↦ {𝑣𝐵𝑢 ∈ (𝑔𝑣)})))
4516, 44eqtrid 2786 . . 3 (𝜑𝐻 = (𝑔 ∈ (𝒫 𝐴m 𝐵) ↦ (𝑢𝐴 ↦ {𝑣𝐵𝑢 ∈ (𝑔𝑣)})))
46 fveq1 6905 . . . . . 6 (𝑔 = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) → (𝑔𝑣) = ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣))
4746eleq2d 2824 . . . . 5 (𝑔 = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) → (𝑢 ∈ (𝑔𝑣) ↔ 𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)))
4847rabbidv 3440 . . . 4 (𝑔 = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) → {𝑣𝐵𝑢 ∈ (𝑔𝑣)} = {𝑣𝐵𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)})
4948mpteq2dv 5249 . . 3 (𝑔 = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) → (𝑢𝐴 ↦ {𝑣𝐵𝑢 ∈ (𝑔𝑣)}) = (𝑢𝐴 ↦ {𝑣𝐵𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)}))
5011, 15, 45, 49fmptco 7148 . 2 (𝜑 → (𝐻𝐺) = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑢𝐴 ↦ {𝑣𝐵𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)})))
51 eqidd 2735 . . . . . . . . . . 11 (((𝜑𝑢𝐴) ∧ 𝑣𝐵) → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}))
52 eleq1w 2821 . . . . . . . . . . . . 13 (𝑦 = 𝑣 → (𝑦 ∈ (𝑓𝑥) ↔ 𝑣 ∈ (𝑓𝑥)))
5352rabbidv 3440 . . . . . . . . . . . 12 (𝑦 = 𝑣 → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} = {𝑥𝐴𝑣 ∈ (𝑓𝑥)})
5453adantl 481 . . . . . . . . . . 11 ((((𝜑𝑢𝐴) ∧ 𝑣𝐵) ∧ 𝑦 = 𝑣) → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} = {𝑥𝐴𝑣 ∈ (𝑓𝑥)})
55 simpr 484 . . . . . . . . . . 11 (((𝜑𝑢𝐴) ∧ 𝑣𝐵) → 𝑣𝐵)
56 rabexg 5342 . . . . . . . . . . . . 13 (𝐴𝑉 → {𝑥𝐴𝑣 ∈ (𝑓𝑥)} ∈ V)
571, 56syl 17 . . . . . . . . . . . 12 (𝜑 → {𝑥𝐴𝑣 ∈ (𝑓𝑥)} ∈ V)
5857ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑢𝐴) ∧ 𝑣𝐵) → {𝑥𝐴𝑣 ∈ (𝑓𝑥)} ∈ V)
5951, 54, 55, 58fvmptd 7022 . . . . . . . . . 10 (((𝜑𝑢𝐴) ∧ 𝑣𝐵) → ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣) = {𝑥𝐴𝑣 ∈ (𝑓𝑥)})
6059eleq2d 2824 . . . . . . . . 9 (((𝜑𝑢𝐴) ∧ 𝑣𝐵) → (𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣) ↔ 𝑢 ∈ {𝑥𝐴𝑣 ∈ (𝑓𝑥)}))
61 fveq2 6906 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (𝑓𝑥) = (𝑓𝑢))
6261eleq2d 2824 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝑣 ∈ (𝑓𝑥) ↔ 𝑣 ∈ (𝑓𝑢)))
6362elrab3 3695 . . . . . . . . . 10 (𝑢𝐴 → (𝑢 ∈ {𝑥𝐴𝑣 ∈ (𝑓𝑥)} ↔ 𝑣 ∈ (𝑓𝑢)))
6463ad2antlr 727 . . . . . . . . 9 (((𝜑𝑢𝐴) ∧ 𝑣𝐵) → (𝑢 ∈ {𝑥𝐴𝑣 ∈ (𝑓𝑥)} ↔ 𝑣 ∈ (𝑓𝑢)))
6560, 64bitrd 279 . . . . . . . 8 (((𝜑𝑢𝐴) ∧ 𝑣𝐵) → (𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣) ↔ 𝑣 ∈ (𝑓𝑢)))
6665rabbidva 3439 . . . . . . 7 ((𝜑𝑢𝐴) → {𝑣𝐵𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)} = {𝑣𝐵𝑣 ∈ (𝑓𝑢)})
6766adantlr 715 . . . . . 6 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) ∧ 𝑢𝐴) → {𝑣𝐵𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)} = {𝑣𝐵𝑣 ∈ (𝑓𝑢)})
68 elmapi 8887 . . . . . . . . . . 11 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
6968ad2antlr 727 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) ∧ 𝑢𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
70 simpr 484 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) ∧ 𝑢𝐴) → 𝑢𝐴)
7169, 70ffvelcdmd 7104 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) ∧ 𝑢𝐴) → (𝑓𝑢) ∈ 𝒫 𝐵)
7271elpwid 4613 . . . . . . . 8 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) ∧ 𝑢𝐴) → (𝑓𝑢) ⊆ 𝐵)
73 sseqin2 4230 . . . . . . . 8 ((𝑓𝑢) ⊆ 𝐵 ↔ (𝐵 ∩ (𝑓𝑢)) = (𝑓𝑢))
7472, 73sylib 218 . . . . . . 7 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) ∧ 𝑢𝐴) → (𝐵 ∩ (𝑓𝑢)) = (𝑓𝑢))
75 dfin5 3970 . . . . . . 7 (𝐵 ∩ (𝑓𝑢)) = {𝑣𝐵𝑣 ∈ (𝑓𝑢)}
7674, 75eqtr3di 2789 . . . . . 6 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) ∧ 𝑢𝐴) → (𝑓𝑢) = {𝑣𝐵𝑣 ∈ (𝑓𝑢)})
7767, 76eqtr4d 2777 . . . . 5 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) ∧ 𝑢𝐴) → {𝑣𝐵𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)} = (𝑓𝑢))
7877mpteq2dva 5247 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) → (𝑢𝐴 ↦ {𝑣𝐵𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)}) = (𝑢𝐴 ↦ (𝑓𝑢)))
7968feqmptd 6976 . . . . 5 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝑓 = (𝑢𝐴 ↦ (𝑓𝑢)))
8079adantl 481 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) → 𝑓 = (𝑢𝐴 ↦ (𝑓𝑢)))
8178, 80eqtr4d 2777 . . 3 ((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) → (𝑢𝐴 ↦ {𝑣𝐵𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)}) = 𝑓)
8281mpteq2dva 5247 . 2 (𝜑 → (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑢𝐴 ↦ {𝑣𝐵𝑢 ∈ ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})‘𝑣)})) = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ 𝑓))
83 mptresid 6070 . . . 4 ( I ↾ (𝒫 𝐵m 𝐴)) = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ 𝑓)
8483eqcomi 2743 . . 3 (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ 𝑓) = ( I ↾ (𝒫 𝐵m 𝐴))
8584a1i 11 . 2 (𝜑 → (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ 𝑓) = ( I ↾ (𝒫 𝐵m 𝐴)))
8650, 82, 853eqtrd 2778 1 (𝜑 → (𝐻𝐺) = ( I ↾ (𝒫 𝐵m 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  cin 3961  wss 3962  𝒫 cpw 4604  cmpt 5230   I cid 5581  cres 5690  ccom 5692  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  m cmap 8864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866
This theorem is referenced by:  fsovcnvd  44003
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