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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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