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Theorem cnpresti 22784
Description: One direction of cnprest 22785 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 2733 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
31, 2cnpf 22743 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
433ad2ant3 1136 . . 3 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
5 simp1 1137 . . 3 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐴 βŠ† 𝑋)
64, 5fssresd 6756 . 2 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝐹 β†Ύ 𝐴):𝐴⟢βˆͺ 𝐾)
7 simpl2 1193 . . . . . 6 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑦 ∈ 𝐾) β†’ 𝑃 ∈ 𝐴)
87fvresd 6909 . . . . 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 2819 . . . 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 22752 . . . . . . 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 1122 . . . . . 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 1188 . . . . 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 1138 . . . . . . . . . . 11 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝑃 ∈ 𝐴)
1513, 14jctird 528 . . . . . . . . . 10 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝑃 ∈ π‘₯ β†’ (𝑃 ∈ π‘₯ ∧ 𝑃 ∈ 𝐴)))
16 elin 3964 . . . . . . . . . 10 (𝑃 ∈ (π‘₯ ∩ 𝐴) ↔ (𝑃 ∈ π‘₯ ∧ 𝑃 ∈ 𝐴))
1715, 16syl6ibr 252 . . . . . . . . 9 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝑃 ∈ π‘₯ β†’ 𝑃 ∈ (π‘₯ ∩ 𝐴)))
18 inss1 4228 . . . . . . . . . . 11 (π‘₯ ∩ 𝐴) βŠ† π‘₯
19 imass2 6099 . . . . . . . . . . 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 3993 . . . . . . . . 9 ((𝐹 β€œ π‘₯) βŠ† 𝑦 β†’ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)
2317, 22anim12d1 611 . . . . . . . 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 3171 . . . . . . 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 3479 . . . . . . . . . 10 π‘₯ ∈ V
2625inex1 5317 . . . . . . . . 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 22738 . . . . . . . . . 10 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) β†’ 𝐽 ∈ Top)
29283ad2ant3 1136 . . . . . . . . 9 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐽 ∈ Top)
3029uniexd 7729 . . . . . . . . . 10 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ βˆͺ 𝐽 ∈ V)
315, 1sseqtrdi 4032 . . . . . . . . . 10 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐴 βŠ† βˆͺ 𝐽)
3230, 31ssexd 5324 . . . . . . . . 9 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐴 ∈ V)
33 elrest 17370 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) β†’ (𝑧 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑧 = (π‘₯ ∩ 𝐴)))
3429, 32, 33syl2anc 585 . . . . . . . 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 486 . . . . . . . . . 10 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ 𝑧 = (π‘₯ ∩ 𝐴))
3635eleq2d 2820 . . . . . . . . 9 (((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ (π‘₯ ∩ 𝐴)))
3735imaeq2d 6058 . . . . . . . . . . 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 4229 . . . . . . . . . . . 12 (π‘₯ ∩ 𝐴) βŠ† 𝐴
39 resima2 6015 . . . . . . . . . . . 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 2789 . . . . . . . . . 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 4013 . . . . . . . . 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 632 . . . . . . . 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 5409 . . . . . . 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 482 . . . . 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 3147 . 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 22411 . . . . 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 22657 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄))
5351, 5, 52syl2anc 585 . . 3 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄))
54 cnptop2 22739 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) β†’ 𝐾 ∈ Top)
55543ad2ant3 1136 . . . 4 ((𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐾 ∈ Top)
562toptopon 22411 . . . 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 22733 . . 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 1372 . 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 712 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   ∩ cin 3947   βŠ† wss 3948  βˆͺ cuni 4908   β†Ύ cres 5678   β€œ cima 5679  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406   β†Ύt crest 17363  Topctop 22387  TopOnctopon 22404   CnP ccnp 22721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-map 8819  df-en 8937  df-fin 8940  df-fi 9403  df-rest 17365  df-topgen 17386  df-top 22388  df-topon 22405  df-bases 22441  df-cnp 22724
This theorem is referenced by:  efrlim  26464  cvmlift2lem11  34293
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