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Theorem enprmaplem3 6078
 Description: Lemma for enprmap 6082. The converse of W is a function. (Contributed by SF, 3-Mar-2015.)
Hypothesis
Ref Expression
enprmaplem3.1 W = (r (Am B) (r “ {x}))
Assertion
Ref Expression
enprmaplem3 ((xy A = {x, y}) → Fun W)
Distinct variable groups:   A,r   B,r   x,r
Allowed substitution hints:   A(x,y)   B(x,y)   W(x,y,r)

Proof of Theorem enprmaplem3
Dummy variables p q w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brcnv 4892 . . . . . 6 (zWppWz)
2 brcnv 4892 . . . . . 6 (zWqqWz)
3 breldm 4911 . . . . . . . . 9 (pWzp dom W)
4 enprmaplem3.1 . . . . . . . . . . 11 W = (r (Am B) (r “ {x}))
54enprmaplem2 6077 . . . . . . . . . 10 W Fn (Am B)
6 fndm 5182 . . . . . . . . . 10 (W Fn (Am B) → dom W = (Am B))
75, 6ax-mp 5 . . . . . . . . 9 dom W = (Am B)
83, 7syl6eleq 2443 . . . . . . . 8 (pWzp (Am B))
9 fnfun 5181 . . . . . . . . . . 11 (W Fn (Am B) → Fun W)
105, 9ax-mp 5 . . . . . . . . . 10 Fun W
11 funbrfv 5356 . . . . . . . . . 10 (Fun W → (pWz → (Wp) = z))
1210, 11ax-mp 5 . . . . . . . . 9 (pWz → (Wp) = z)
13 cnveq 4886 . . . . . . . . . . . 12 (r = pr = p)
1413imaeq1d 4941 . . . . . . . . . . 11 (r = p → (r “ {x}) = (p “ {x}))
15 vex 2862 . . . . . . . . . . . . 13 p V
1615cnvex 5102 . . . . . . . . . . . 12 p V
17 snex 4111 . . . . . . . . . . . 12 {x} V
1816, 17imaex 4747 . . . . . . . . . . 11 (p “ {x}) V
1914, 4, 18fvmpt 5700 . . . . . . . . . 10 (p (Am B) → (Wp) = (p “ {x}))
208, 19syl 15 . . . . . . . . 9 (pWz → (Wp) = (p “ {x}))
2112, 20eqtr3d 2387 . . . . . . . 8 (pWzz = (p “ {x}))
228, 21jca 518 . . . . . . 7 (pWz → (p (Am B) z = (p “ {x})))
23 breldm 4911 . . . . . . . . 9 (qWzq dom W)
2423, 7syl6eleq 2443 . . . . . . . 8 (qWzq (Am B))
25 funbrfv 5356 . . . . . . . . . 10 (Fun W → (qWz → (Wq) = z))
2610, 25ax-mp 5 . . . . . . . . 9 (qWz → (Wq) = z)
27 cnveq 4886 . . . . . . . . . . . 12 (r = qr = q)
2827imaeq1d 4941 . . . . . . . . . . 11 (r = q → (r “ {x}) = (q “ {x}))
29 vex 2862 . . . . . . . . . . . . 13 q V
3029cnvex 5102 . . . . . . . . . . . 12 q V
3130, 17imaex 4747 . . . . . . . . . . 11 (q “ {x}) V
3228, 4, 31fvmpt 5700 . . . . . . . . . 10 (q (Am B) → (Wq) = (q “ {x}))
3324, 32syl 15 . . . . . . . . 9 (qWz → (Wq) = (q “ {x}))
3426, 33eqtr3d 2387 . . . . . . . 8 (qWzz = (q “ {x}))
3524, 34jca 518 . . . . . . 7 (qWz → (q (Am B) z = (q “ {x})))
3622, 35anim12i 549 . . . . . 6 ((pWz qWz) → ((p (Am B) z = (p “ {x})) (q (Am B) z = (q “ {x}))))
371, 2, 36syl2anb 465 . . . . 5 ((zWp zWq) → ((p (Am B) z = (p “ {x})) (q (Am B) z = (q “ {x}))))
38 elmapi 6016 . . . . . . . . 9 (p (Am B) → p:B–→A)
39 elmapi 6016 . . . . . . . . 9 (q (Am B) → q:B–→A)
4038, 39anim12i 549 . . . . . . . 8 ((p (Am B) q (Am B)) → (p:B–→A q:B–→A))
41 eqtr2 2371 . . . . . . . 8 ((z = (p “ {x}) z = (q “ {x})) → (p “ {x}) = (q “ {x}))
42 simprll 738 . . . . . . . . . . 11 (((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) → p:B–→A)
43 ffn 5223 . . . . . . . . . . 11 (p:B–→Ap Fn B)
4442, 43syl 15 . . . . . . . . . 10 (((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) → p Fn B)
45 simprlr 739 . . . . . . . . . . 11 (((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) → q:B–→A)
46 ffn 5223 . . . . . . . . . . 11 (q:B–→Aq Fn B)
4745, 46syl 15 . . . . . . . . . 10 (((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) → q Fn B)
48 ffvelrn 5415 . . . . . . . . . . . 12 ((p:B–→A z B) → (pz) A)
4942, 48sylan 457 . . . . . . . . . . 11 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (pz) A)
50 simpllr 735 . . . . . . . . . . . . . 14 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → A = {x, y})
5150eleq2d 2420 . . . . . . . . . . . . 13 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((pz) A ↔ (pz) {x, y}))
52 fvex 5339 . . . . . . . . . . . . . 14 (pz) V
5352elpr 3751 . . . . . . . . . . . . 13 ((pz) {x, y} ↔ ((pz) = x (pz) = y))
5451, 53syl6bb 252 . . . . . . . . . . . 12 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((pz) A ↔ ((pz) = x (pz) = y)))
55 simprr 733 . . . . . . . . . . . . . . 15 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B (pz) = x)) → (pz) = x)
56 simplrr 737 . . . . . . . . . . . . . . . . . . . 20 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (p “ {x}) = (q “ {x}))
5756eleq2d 2420 . . . . . . . . . . . . . . . . . . 19 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (z (p “ {x}) ↔ z (q “ {x})))
58 eliniseg 5020 . . . . . . . . . . . . . . . . . . 19 (z (p “ {x}) ↔ zpx)
59 eliniseg 5020 . . . . . . . . . . . . . . . . . . 19 (z (q “ {x}) ↔ zqx)
6057, 58, 593bitr3g 278 . . . . . . . . . . . . . . . . . 18 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (zpxzqx))
6160biimpd 198 . . . . . . . . . . . . . . . . 17 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (zpxzqx))
62 fnbrfvb 5358 . . . . . . . . . . . . . . . . . 18 ((p Fn B z B) → ((pz) = xzpx))
6344, 62sylan 457 . . . . . . . . . . . . . . . . 17 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((pz) = xzpx))
64 fnbrfvb 5358 . . . . . . . . . . . . . . . . . 18 ((q Fn B z B) → ((qz) = xzqx))
6547, 64sylan 457 . . . . . . . . . . . . . . . . 17 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((qz) = xzqx))
6661, 63, 653imtr4d 259 . . . . . . . . . . . . . . . 16 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((pz) = x → (qz) = x))
6766impr 602 . . . . . . . . . . . . . . 15 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B (pz) = x)) → (qz) = x)
6855, 67eqtr4d 2388 . . . . . . . . . . . . . 14 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B (pz) = x)) → (pz) = (qz))
6968expr 598 . . . . . . . . . . . . 13 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((pz) = x → (pz) = (qz)))
70 simprr 733 . . . . . . . . . . . . . . 15 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B (pz) = y)) → (pz) = y)
71 simplll 734 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → xy)
7271neneqd 2532 . . . . . . . . . . . . . . . . . . . . . . 23 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → ¬ x = y)
7342adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → p:B–→A)
74 ffun 5225 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (p:B–→A → Fun p)
75 fununiq 5517 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun p zpx zpy) → x = y)
76753expib 1154 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Fun p → ((zpx zpy) → x = y))
7776ancomsd 440 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Fun p → ((zpy zpx) → x = y))
7873, 74, 773syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((zpy zpx) → x = y))
7978exp3a 425 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (zpy → (zpxx = y)))
8079impr 602 . . . . . . . . . . . . . . . . . . . . . . 23 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (zpxx = y))
8172, 80mtod 168 . . . . . . . . . . . . . . . . . . . . . 22 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → ¬ zpx)
8281expr 598 . . . . . . . . . . . . . . . . . . . . 21 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (zpy → ¬ zpx))
8360biimprd 214 . . . . . . . . . . . . . . . . . . . . 21 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (zqxzpx))
8482, 83nsyld 132 . . . . . . . . . . . . . . . . . . . 20 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (zpy → ¬ zqx))
8584impr 602 . . . . . . . . . . . . . . . . . . 19 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → ¬ zqx)
86 simprl 732 . . . . . . . . . . . . . . . . . . . . 21 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → z B)
8745adantr 451 . . . . . . . . . . . . . . . . . . . . . 22 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → q:B–→A)
88 fdm 5226 . . . . . . . . . . . . . . . . . . . . . 22 (q:B–→A → dom q = B)
8987, 88syl 15 . . . . . . . . . . . . . . . . . . . . 21 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → dom q = B)
9086, 89eleqtrrd 2430 . . . . . . . . . . . . . . . . . . . 20 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → z dom q)
91 eldm 4898 . . . . . . . . . . . . . . . . . . . . 21 (z dom qw zqw)
92 brelrn 4960 . . . . . . . . . . . . . . . . . . . . . . . 24 (zqww ran q)
93 frn 5228 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (q:B–→A → ran q A)
9487, 93syl 15 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → ran q A)
9594sseld 3272 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (w ran qw A))
9692, 95syl5 28 . . . . . . . . . . . . . . . . . . . . . . 23 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (zqww A))
97 simpllr 735 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → A = {x, y})
9897eleq2d 2420 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (w Aw {x, y}))
99 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 w V
10099elpr 3751 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (w {x, y} ↔ (w = x w = y))
101 breq2 4643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (w = x → (zqwzqx))
102101biimpcd 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (zqw → (w = xzqx))
103 breq2 4643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (w = y → (zqwzqy))
104103biimpcd 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (zqw → (w = yzqy))
105102, 104orim12d 811 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (zqw → ((w = x w = y) → (zqx zqy)))
106105com12 27 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((w = x w = y) → (zqw → (zqx zqy)))
107100, 106sylbi 187 . . . . . . . . . . . . . . . . . . . . . . . . 25 (w {x, y} → (zqw → (zqx zqy)))
10898, 107syl6bi 219 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (w A → (zqw → (zqx zqy))))
109108com23 72 . . . . . . . . . . . . . . . . . . . . . . 23 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (zqw → (w A → (zqx zqy))))
11096, 109mpdd 36 . . . . . . . . . . . . . . . . . . . . . 22 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (zqw → (zqx zqy)))
111110exlimdv 1636 . . . . . . . . . . . . . . . . . . . . 21 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (w zqw → (zqx zqy)))
11291, 111syl5bi 208 . . . . . . . . . . . . . . . . . . . 20 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (z dom q → (zqx zqy)))
11390, 112mpd 14 . . . . . . . . . . . . . . . . . . 19 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → (zqx zqy))
114 orel1 371 . . . . . . . . . . . . . . . . . . 19 zqx → ((zqx zqy) → zqy))
11585, 113, 114sylc 56 . . . . . . . . . . . . . . . . . 18 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B zpy)) → zqy)
116115expr 598 . . . . . . . . . . . . . . . . 17 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (zpyzqy))
117 fnbrfvb 5358 . . . . . . . . . . . . . . . . . 18 ((p Fn B z B) → ((pz) = yzpy))
11844, 117sylan 457 . . . . . . . . . . . . . . . . 17 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((pz) = yzpy))
119 fnbrfvb 5358 . . . . . . . . . . . . . . . . . 18 ((q Fn B z B) → ((qz) = yzqy))
12047, 119sylan 457 . . . . . . . . . . . . . . . . 17 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((qz) = yzqy))
121116, 118, 1203imtr4d 259 . . . . . . . . . . . . . . . 16 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((pz) = y → (qz) = y))
122121impr 602 . . . . . . . . . . . . . . 15 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B (pz) = y)) → (qz) = y)
12370, 122eqtr4d 2388 . . . . . . . . . . . . . 14 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) (z B (pz) = y)) → (pz) = (qz))
124123expr 598 . . . . . . . . . . . . 13 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((pz) = y → (pz) = (qz)))
12569, 124jaod 369 . . . . . . . . . . . 12 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (((pz) = x (pz) = y) → (pz) = (qz)))
12654, 125sylbid 206 . . . . . . . . . . 11 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → ((pz) A → (pz) = (qz)))
12749, 126mpd 14 . . . . . . . . . 10 ((((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) z B) → (pz) = (qz))
12844, 47, 127eqfnfvd 5395 . . . . . . . . 9 (((xy A = {x, y}) ((p:B–→A q:B–→A) (p “ {x}) = (q “ {x}))) → p = q)
129128expcom 424 . . . . . . . 8 (((p:B–→A q:B–→A) (p “ {x}) = (q “ {x})) → ((xy A = {x, y}) → p = q))
13040, 41, 129syl2an 463 . . . . . . 7 (((p (Am B) q (Am B)) (z = (p “ {x}) z = (q “ {x}))) → ((xy A = {x, y}) → p = q))
131130an4s 799 . . . . . 6 (((p (Am B) z = (p “ {x})) (q (Am B) z = (q “ {x}))) → ((xy A = {x, y}) → p = q))
132131com12 27 . . . . 5 ((xy A = {x, y}) → (((p (Am B) z = (p “ {x})) (q (Am B) z = (q “ {x}))) → p = q))
13337, 132syl5 28 . . . 4 ((xy A = {x, y}) → ((zWp zWq) → p = q))
134133alrimiv 1631 . . 3 ((xy A = {x, y}) → q((zWp zWq) → p = q))
135134alrimivv 1632 . 2 ((xy A = {x, y}) → zpq((zWp zWq) → p = q))
136 dffun2 5119 . 2 (Fun Wzpq((zWp zWq) → p = q))
137135, 136sylibr 203 1 ((xy A = {x, y}) → Fun W)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516   ⊆ wss 3257  {csn 3737  {cpr 3738   class class class wbr 4639   “ cima 4722  ◡ccnv 4771  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777   ‘cfv 4781  (class class class)co 5525   ↦ cmpt 5651   ↑m cmap 5999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001 This theorem is referenced by:  enprmap  6082
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