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Theorem iscnp3 21849
 Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". (Contributed by NM, 15-May-2007.)
Assertion
Ref Expression
iscnp3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑃,𝑦

Proof of Theorem iscnp3
StepHypRef Expression
1 iscnp 21842 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
2 ffun 6490 . . . . . . . . . 10 (𝐹:𝑋𝑌 → Fun 𝐹)
32ad2antlr 726 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → Fun 𝐹)
4 toponss 21532 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
54adantlr 714 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → 𝑥𝑋)
6 fdm 6495 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
76ad2antlr 726 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → dom 𝐹 = 𝑋)
85, 7sseqtrrd 3956 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → 𝑥 ⊆ dom 𝐹)
9 funimass3 6801 . . . . . . . . 9 ((Fun 𝐹𝑥 ⊆ dom 𝐹) → ((𝐹𝑥) ⊆ 𝑦𝑥 ⊆ (𝐹𝑦)))
103, 8, 9syl2anc 587 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → ((𝐹𝑥) ⊆ 𝑦𝑥 ⊆ (𝐹𝑦)))
1110anbi2d 631 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))
1211rexbidva 3255 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))
1312imbi2d 344 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) → (((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦)))))
1413ralbidv 3162 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦)))))
1514pm5.32da 582 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))))
16153ad2ant1 1130 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → ((𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))))
171, 16bitrd 282 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ◡ccnv 5518  dom cdm 5519   “ cima 5522  Fun wfun 6318  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  TopOnctopon 21515   CnP ccnp 21830 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-top 21499  df-topon 21516  df-cnp 21833 This theorem is referenced by:  cncnpi  21883  cnpdis  21898
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