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Theorem cnpflf2 23725
Description: 𝐹 is continuous at point 𝐴 iff a limit of 𝐹 when π‘₯ tends to 𝐴 is (πΉβ€˜π΄). Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
cnpflf2.3 𝐿 = ((neiβ€˜π½)β€˜{𝐴})
Assertion
Ref Expression
cnpflf2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ))))
Proof of Theorem cnpflf2
Dummy variables 𝑒 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnpf2 22975 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expa 1117 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
323adantl3 1167 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
4 simpl1 1190 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
5 simpl3 1192 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐴 ∈ 𝑋)
6 neiflim 23699 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})))
7 cnpflf2.3 . . . . . . 7 𝐿 = ((neiβ€˜π½)β€˜{𝐴})
87oveq2i 7423 . . . . . 6 (𝐽 fLim 𝐿) = (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴}))
96, 8eleqtrrdi 2843 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ (𝐽 fLim 𝐿))
104, 5, 9syl2anc 583 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐴 ∈ (𝐽 fLim 𝐿))
11 simpr 484 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄))
12 cnpflfi 23724 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ))
1310, 11, 12syl2anc 583 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ))
143, 13jca 511 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ)))
15 simpl1 1190 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
16 topontop 22636 . . . . . . . . . . . 12 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
1715, 16syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐽 ∈ Top)
18 simpl3 1192 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐴 ∈ 𝑋)
19 toponuni 22637 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2015, 19syl 17 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝑋 = βˆͺ 𝐽)
2118, 20eleqtrd 2834 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐽)
227eleq2i 2824 . . . . . . . . . . . 12 (𝑧 ∈ 𝐿 ↔ 𝑧 ∈ ((neiβ€˜π½)β€˜{𝐴}))
23 eqid 2731 . . . . . . . . . . . . 13 βˆͺ 𝐽 = βˆͺ 𝐽
2423isneip 22830 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴 ∈ βˆͺ 𝐽) β†’ (𝑧 ∈ ((neiβ€˜π½)β€˜{𝐴}) ↔ (𝑧 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧))))
2522, 24bitrid 283 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴 ∈ βˆͺ 𝐽) β†’ (𝑧 ∈ 𝐿 ↔ (𝑧 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧))))
2617, 21, 25syl2anc 583 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (𝑧 ∈ 𝐿 ↔ (𝑧 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧))))
27 sstr2 3989 . . . . . . . . . . . . . . 15 ((𝐹 β€œ 𝑣) βŠ† (𝐹 β€œ 𝑧) β†’ ((𝐹 β€œ 𝑧) βŠ† 𝑒 β†’ (𝐹 β€œ 𝑣) βŠ† 𝑒))
28 imass2 6101 . . . . . . . . . . . . . . 15 (𝑣 βŠ† 𝑧 β†’ (𝐹 β€œ 𝑣) βŠ† (𝐹 β€œ 𝑧))
2927, 28syl11 33 . . . . . . . . . . . . . 14 ((𝐹 β€œ 𝑧) βŠ† 𝑒 β†’ (𝑣 βŠ† 𝑧 β†’ (𝐹 β€œ 𝑣) βŠ† 𝑒))
3029anim2d 611 . . . . . . . . . . . . 13 ((𝐹 β€œ 𝑧) βŠ† 𝑒 β†’ ((𝐴 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧) β†’ (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒)))
3130reximdv 3169 . . . . . . . . . . . 12 ((𝐹 β€œ 𝑧) βŠ† 𝑒 β†’ (βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧) β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒)))
3231com12 32 . . . . . . . . . . 11 (βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧) β†’ ((𝐹 β€œ 𝑧) βŠ† 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒)))
3332adantl 481 . . . . . . . . . 10 ((𝑧 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧)) β†’ ((𝐹 β€œ 𝑧) βŠ† 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒)))
3426, 33syl6bi 253 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (𝑧 ∈ 𝐿 β†’ ((𝐹 β€œ 𝑧) βŠ† 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒))))
3534rexlimdv 3152 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆƒπ‘§ ∈ 𝐿 (𝐹 β€œ 𝑧) βŠ† 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒)))
3635imim2d 57 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘§ ∈ 𝐿 (𝐹 β€œ 𝑧) βŠ† 𝑒) β†’ ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒))))
3736ralimdv 3168 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘§ ∈ 𝐿 (𝐹 β€œ 𝑧) βŠ† 𝑒) β†’ βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒))))
38 simpr 484 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
3937, 38jctild 525 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘§ ∈ 𝐿 (𝐹 β€œ 𝑧) βŠ† 𝑒) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒)))))
4039adantld 490 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (((πΉβ€˜π΄) ∈ π‘Œ ∧ βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘§ ∈ 𝐿 (𝐹 β€œ 𝑧) βŠ† 𝑒)) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒)))))
41 simpl2 1191 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
4218snssd 4812 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ {𝐴} βŠ† 𝑋)
4318snn0d 4779 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ {𝐴} β‰  βˆ…)
44 neifil 23605 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ {𝐴} βŠ† 𝑋 ∧ {𝐴} β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
4515, 42, 43, 44syl3anc 1370 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
467, 45eqeltrid 2836 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐿 ∈ (Filβ€˜π‘‹))
47 isflf 23718 . . . . 5 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 ∈ (Filβ€˜π‘‹) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ) ↔ ((πΉβ€˜π΄) ∈ π‘Œ ∧ βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘§ ∈ 𝐿 (𝐹 β€œ 𝑧) βŠ† 𝑒))))
4841, 46, 38, 47syl3anc 1370 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ) ↔ ((πΉβ€˜π΄) ∈ π‘Œ ∧ βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘§ ∈ 𝐿 (𝐹 β€œ 𝑧) βŠ† 𝑒))))
49 iscnp 22962 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒)))))
5049adantr 480 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘’ ∈ 𝐾 ((πΉβ€˜π΄) ∈ 𝑒 β†’ βˆƒπ‘£ ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 β€œ 𝑣) βŠ† 𝑒)))))
5140, 48, 503imtr4d 294 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)))
5251impr 454 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ))) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄))
5314, 52impbida 798 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βˆͺ cuni 4908   β€œ cima 5679  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412  Topctop 22616  TopOnctopon 22633  neicnei 22822   CnP ccnp 22950  Filcfil 23570   fLim cflim 23659   fLimf cflf 23660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-map 8825  df-fbas 21142  df-fg 21143  df-top 22617  df-topon 22634  df-ntr 22745  df-nei 22823  df-cnp 22953  df-fil 23571  df-fm 23663  df-flim 23664  df-flf 23665
This theorem is referenced by:  cnpflf  23726
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