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Theorem cnpresti 23113
Description: One direction of cnprest 23114 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 1-May-2015.)
Hypothesis
Ref Expression
cnprest.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
cnpresti ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ))
Proof of Theorem cnpresti
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnprest.1 . . . . 5 𝑋 = βˆͺ 𝐽
2 eqid 2724 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
31, 2cnpf 23072 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
433ad2ant3 1132 . . 3 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
5 simp1 1133 . . 3 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐴 βŠ† 𝑋)
64, 5fssresd 6748 . 2 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝐹 β†Ύ 𝐴):𝐴⟢βˆͺ 𝐾)
7 simpl2 1189 . . . . . 6 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ 𝐾) β†’ 𝑃 ∈ 𝐴)
87fvresd 6901 . . . . 5 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ 𝐾) β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) = (πΉβ€˜π‘ƒ))
98eleq1d 2810 . . . 4 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ 𝐾) β†’ (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 ↔ (πΉβ€˜π‘ƒ) ∈ 𝑦))
10 cnpimaex 23081 . . . . . . 7 ((𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦))
11103expia 1118 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝑦 ∈ 𝐾) β†’ ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))
12113ad2antl3 1184 . . . . 5 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ 𝐾) β†’ ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))
13 idd 24 . . . . . . . . . . 11 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝑃 ∈ π‘₯ β†’ 𝑃 ∈ π‘₯))
14 simp2 1134 . . . . . . . . . . 11 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝑃 ∈ 𝐴)
1513, 14jctird 526 . . . . . . . . . 10 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝑃 ∈ π‘₯ β†’ (𝑃 ∈ π‘₯ ∧ 𝑃 ∈ 𝐴)))
16 elin 3956 . . . . . . . . . 10 (𝑃 ∈ (π‘₯ ∩ 𝐴) ↔ (𝑃 ∈ π‘₯ ∧ 𝑃 ∈ 𝐴))
1715, 16imbitrrdi 251 . . . . . . . . 9 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝑃 ∈ π‘₯ β†’ 𝑃 ∈ (π‘₯ ∩ 𝐴)))
18 inss1 4220 . . . . . . . . . . 11 (π‘₯ ∩ 𝐴) βŠ† π‘₯
19 imass2 6091 . . . . . . . . . . 11 ((π‘₯ ∩ 𝐴) βŠ† π‘₯ β†’ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† (𝐹 β€œ π‘₯))
2018, 19ax-mp 5 . . . . . . . . . 10 (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† (𝐹 β€œ π‘₯)
21 id 22 . . . . . . . . . 10 ((𝐹 β€œ π‘₯) βŠ† 𝑦 β†’ (𝐹 β€œ π‘₯) βŠ† 𝑦)
2220, 21sstrid 3985 . . . . . . . . 9 ((𝐹 β€œ π‘₯) βŠ† 𝑦 β†’ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)
2317, 22anim12d1 609 . . . . . . . 8 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ ((𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ (𝑃 ∈ (π‘₯ ∩ 𝐴) ∧ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)))
2423reximdv 3162 . . . . . . 7 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ (π‘₯ ∩ 𝐴) ∧ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)))
25 vex 3470 . . . . . . . . . 10 π‘₯ ∈ V
2625inex1 5307 . . . . . . . . 9 (π‘₯ ∩ 𝐴) ∈ V
2726a1i 11 . . . . . . . 8 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ π‘₯ ∈ 𝐽) β†’ (π‘₯ ∩ 𝐴) ∈ V)
28 cnptop1 23067 . . . . . . . . . 10 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) β†’ 𝐽 ∈ Top)
29283ad2ant3 1132 . . . . . . . . 9 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐽 ∈ Top)
3029uniexd 7725 . . . . . . . . . 10 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ βˆͺ 𝐽 ∈ V)
315, 1sseqtrdi 4024 . . . . . . . . . 10 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐴 βŠ† βˆͺ 𝐽)
3230, 31ssexd 5314 . . . . . . . . 9 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐴 ∈ V)
33 elrest 17371 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) β†’ (𝑧 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑧 = (π‘₯ ∩ 𝐴)))
3429, 32, 33syl2anc 583 . . . . . . . 8 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝑧 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑧 = (π‘₯ ∩ 𝐴)))
35 simpr 484 . . . . . . . . . 10 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ 𝑧 = (π‘₯ ∩ 𝐴))
3635eleq2d 2811 . . . . . . . . 9 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ (π‘₯ ∩ 𝐴)))
3735imaeq2d 6049 . . . . . . . . . . 11 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) = ((𝐹 β†Ύ 𝐴) β€œ (π‘₯ ∩ 𝐴)))
38 inss2 4221 . . . . . . . . . . . 12 (π‘₯ ∩ 𝐴) βŠ† 𝐴
39 resima2 6006 . . . . . . . . . . . 12 ((π‘₯ ∩ 𝐴) βŠ† 𝐴 β†’ ((𝐹 β†Ύ 𝐴) β€œ (π‘₯ ∩ 𝐴)) = (𝐹 β€œ (π‘₯ ∩ 𝐴)))
4038, 39ax-mp 5 . . . . . . . . . . 11 ((𝐹 β†Ύ 𝐴) β€œ (π‘₯ ∩ 𝐴)) = (𝐹 β€œ (π‘₯ ∩ 𝐴))
4137, 40eqtrdi 2780 . . . . . . . . . 10 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) = (𝐹 β€œ (π‘₯ ∩ 𝐴)))
4241sseq1d 4005 . . . . . . . . 9 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ (((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦 ↔ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦))
4336, 42anbi12d 630 . . . . . . . 8 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ ((𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦) ↔ (𝑃 ∈ (π‘₯ ∩ 𝐴) ∧ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)))
4427, 34, 43rexxfr2d 5399 . . . . . . 7 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦) ↔ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ (π‘₯ ∩ 𝐴) ∧ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)))
4524, 44sylibrd 259 . . . . . 6 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
4645adantr 480 . . . . 5 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ 𝐾) β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
4712, 46syld 47 . . . 4 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ 𝐾) β†’ ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
489, 47sylbid 239 . . 3 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ 𝐾) β†’ (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
4948ralrimiva 3138 . 2 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ βˆ€π‘¦ ∈ 𝐾 (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
501toptopon 22740 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
5129, 50sylib 217 . . . 4 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
52 resttopon 22986 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄))
5351, 5, 52syl2anc 583 . . 3 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄))
54 cnptop2 23068 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) β†’ 𝐾 ∈ Top)
55543ad2ant3 1132 . . . 4 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐾 ∈ Top)
562toptopon 22740 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
5755, 56sylib 217 . . 3 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
58 iscnp 23062 . . 3 (((𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝑃 ∈ 𝐴) β†’ ((𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ) ↔ ((𝐹 β†Ύ 𝐴):𝐴⟢βˆͺ 𝐾 ∧ βˆ€π‘¦ ∈ 𝐾 (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))))
5953, 57, 14, 58syl3anc 1368 . 2 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ ((𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ) ↔ ((𝐹 β†Ύ 𝐴):𝐴⟢βˆͺ 𝐾 ∧ βˆ€π‘¦ ∈ 𝐾 (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))))
606, 49, 59mpbir2and 710 1 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062  Vcvv 3466   ∩ cin 3939   βŠ† wss 3940  βˆͺ cuni 4899   β†Ύ cres 5668   β€œ cima 5669  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401   β†Ύt crest 17364  Topctop 22716  TopOnctopon 22733   CnP ccnp 23050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-map 8817  df-en 8935  df-fin 8938  df-fi 9401  df-rest 17366  df-topgen 17387  df-top 22717  df-topon 22734  df-bases 22770  df-cnp 23053
This theorem is referenced by:  efrlim  26816  efrlimOLD  26817  cvmlift2lem11  34759
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