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Theorem cncnp 12436
 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 12403 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
21simprbda 381 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
3 eqid 2140 . . . . . . 7 𝐽 = 𝐽
43cncnpi 12434 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
54ralrimiva 2508 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
65adantl 275 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
7 toponuni 12219 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
87ad2antrr 480 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑋 = 𝐽)
98raleqdv 2635 . . . 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 166 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
112, 10jca 304 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
12 simprl 521 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹:𝑋𝑌)
13 cnvimass 4909 . . . . . . . . . 10 (𝐹𝑦) ⊆ dom 𝐹
14 fdm 5285 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1514adantl 275 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → dom 𝐹 = 𝑋)
1613, 15sseqtrid 3151 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → (𝐹𝑦) ⊆ 𝑋)
17 ssralv 3165 . . . . . . . . 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 531 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
20 simp-4r 532 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐾 ∈ (TopOn‘𝑌))
21 topontop 12218 . . . . . . . . . . . . . 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 522 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
24 cnprcl2k 12412 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝑥𝑋)
2519, 22, 23, 24syl3anc 1217 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑥𝑋)
26 simpllr 524 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑦𝐾)
27 ffn 5279 . . . . . . . . . . . . . 14 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
2827ad2antlr 481 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 Fn 𝑋)
29 simprl 521 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑥 ∈ (𝐹𝑦))
30 elpreima 5546 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑋 → (𝑥 ∈ (𝐹𝑦) ↔ (𝑥𝑋 ∧ (𝐹𝑥) ∈ 𝑦)))
3130simplbda 382 . . . . . . . . . . . . 13 ((𝐹 Fn 𝑋𝑥 ∈ (𝐹𝑦)) → (𝐹𝑥) ∈ 𝑦)
3228, 29, 31syl2anc 409 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (𝐹𝑥) ∈ 𝑦)
33 icnpimaex 12417 . . . . . . . . . . . 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 1232 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
35 simpllr 524 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → 𝐹:𝑋𝑌)
3635ffund 5283 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → Fun 𝐹)
37 toponss 12230 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢𝐽) → 𝑢𝑋)
3819, 37sylan 281 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → 𝑢𝑋)
3935fdmd 5286 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → dom 𝐹 = 𝑋)
4038, 39sseqtrrd 3140 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → 𝑢 ⊆ dom 𝐹)
41 funimass3 5543 . . . . . . . . . . . . . 14 ((Fun 𝐹𝑢 ⊆ dom 𝐹) → ((𝐹𝑢) ⊆ 𝑦𝑢 ⊆ (𝐹𝑦)))
4236, 40, 41syl2anc 409 . . . . . . . . . . . . 13 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → ((𝐹𝑢) ⊆ 𝑦𝑢 ⊆ (𝐹𝑦)))
4342anbi2d 460 . . . . . . . . . . . 12 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → ((𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦) ↔ (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4443rexbidva 2435 . . . . . . . . . . 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 146 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦)))
4645expr 373 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ 𝑥 ∈ (𝐹𝑦)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4746ralimdva 2502 . . . . . . . 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 377 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦)))
5049an32s 558 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦)))
51 topontop 12218 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
5251ad3antrrr 484 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → 𝐽 ∈ Top)
53 eltop2 12276 . . . . . 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 166 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
5655ralrimiva 2508 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)
571adantr 274 . . 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 929 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾))
5911, 58impbida 586 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 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418   ⊆ wss 3075  ∪ cuni 3743  ◡ccnv 4545  dom cdm 4546   “ cima 4549  Fun wfun 5124   Fn wfn 5125  ⟶wf 5126  ‘cfv 5130  (class class class)co 5781  Topctop 12201  TopOnctopon 12214   Cn ccn 12391   CnP ccnp 12392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-topgen 12178  df-top 12202  df-topon 12215  df-cn 12394  df-cnp 12395 This theorem is referenced by:  cncnp2m  12437  cnnei  12438  cnconst2  12439  metcn  12720  txmetcn  12725  cnlimcim  12846  cnlimc  12847  dvcn  12870
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