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Theorem cncnp 14182
Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cncnp ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑥,𝑌

Proof of Theorem cncnp
Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscn 14149 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
21simprbda 383 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
3 eqid 2189 . . . . . . 7 𝐽 = 𝐽
43cncnpi 14180 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
54ralrimiva 2563 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
65adantl 277 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
7 toponuni 13967 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
87ad2antrr 488 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑋 = 𝐽)
98raleqdv 2692 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑥 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
106, 9mpbird 167 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
112, 10jca 306 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
12 simprl 529 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹:𝑋𝑌)
13 cnvimass 5009 . . . . . . . . . 10 (𝐹𝑦) ⊆ dom 𝐹
14 fdm 5390 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1514adantl 277 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → dom 𝐹 = 𝑋)
1613, 15sseqtrid 3220 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → (𝐹𝑦) ⊆ 𝑋)
17 ssralv 3234 . . . . . . . . 9 ((𝐹𝑦) ⊆ 𝑋 → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (𝐹𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
1816, 17syl 14 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (𝐹𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
19 simp-4l 541 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
20 simp-4r 542 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐾 ∈ (TopOn‘𝑌))
21 topontop 13966 . . . . . . . . . . . . . 14 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2220, 21syl 14 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐾 ∈ Top)
23 simprr 531 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
24 cnprcl2k 14158 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝑥𝑋)
2519, 22, 23, 24syl3anc 1249 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑥𝑋)
26 simpllr 534 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑦𝐾)
27 ffn 5384 . . . . . . . . . . . . . 14 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
2827ad2antlr 489 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 Fn 𝑋)
29 simprl 529 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑥 ∈ (𝐹𝑦))
30 elpreima 5655 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑋 → (𝑥 ∈ (𝐹𝑦) ↔ (𝑥𝑋 ∧ (𝐹𝑥) ∈ 𝑦)))
3130simplbda 384 . . . . . . . . . . . . 13 ((𝐹 Fn 𝑋𝑥 ∈ (𝐹𝑦)) → (𝐹𝑥) ∈ 𝑦)
3228, 29, 31syl2anc 411 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (𝐹𝑥) ∈ 𝑦)
33 icnpimaex 14163 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ∧ 𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
3419, 20, 25, 23, 26, 32, 33syl33anc 1264 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
35 simpllr 534 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → 𝐹:𝑋𝑌)
3635ffund 5388 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → Fun 𝐹)
37 toponss 13978 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢𝐽) → 𝑢𝑋)
3819, 37sylan 283 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → 𝑢𝑋)
3935fdmd 5391 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → dom 𝐹 = 𝑋)
4038, 39sseqtrrd 3209 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → 𝑢 ⊆ dom 𝐹)
41 funimass3 5652 . . . . . . . . . . . . . 14 ((Fun 𝐹𝑢 ⊆ dom 𝐹) → ((𝐹𝑢) ⊆ 𝑦𝑢 ⊆ (𝐹𝑦)))
4236, 40, 41syl2anc 411 . . . . . . . . . . . . 13 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → ((𝐹𝑢) ⊆ 𝑦𝑢 ⊆ (𝐹𝑦)))
4342anbi2d 464 . . . . . . . . . . . 12 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → ((𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦) ↔ (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4443rexbidva 2487 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦) ↔ ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4534, 44mpbid 147 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦)))
4645expr 375 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ 𝑥 ∈ (𝐹𝑦)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4746ralimdva 2557 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → (∀𝑥 ∈ (𝐹𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4818, 47syld 45 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4948impr 379 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦)))
5049an32s 568 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦)))
51 topontop 13966 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
5251ad3antrrr 492 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → 𝐽 ∈ Top)
53 eltop2 14022 . . . . . 6 (𝐽 ∈ Top → ((𝐹𝑦) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
5452, 53syl 14 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → ((𝐹𝑦) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
5550, 54mpbird 167 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
5655ralrimiva 2563 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)
571adantr 276 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
5812, 56, 57mpbir2and 946 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾))
5911, 58impbida 596 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wral 2468  wrex 2469  wss 3144   cuni 3824  ccnv 4643  dom cdm 4644  cima 4647  Fun wfun 5229   Fn wfn 5230  wf 5231  cfv 5235  (class class class)co 5895  Topctop 13949  TopOnctopon 13962   Cn ccn 14137   CnP ccnp 14138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-map 6675  df-topgen 12762  df-top 13950  df-topon 13963  df-cn 14140  df-cnp 14141
This theorem is referenced by:  cncnp2m  14183  cnnei  14184  cnconst2  14185  metcn  14466  txmetcn  14471  cnlimcim  14592  cnlimc  14593  dvcn  14616
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