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Theorem iscnp3 23192
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 23185 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
2 ffun 6726 . . . . . . . . . 10 (𝐹:𝑋𝑌 → Fun 𝐹)
32ad2antlr 725 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → Fun 𝐹)
4 toponss 22873 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
54adantlr 713 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → 𝑥𝑋)
6 fdm 6732 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
76ad2antlr 725 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → dom 𝐹 = 𝑋)
85, 7sseqtrrd 4018 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → 𝑥 ⊆ dom 𝐹)
9 funimass3 7062 . . . . . . . . 9 ((Fun 𝐹𝑥 ⊆ dom 𝐹) → ((𝐹𝑥) ⊆ 𝑦𝑥 ⊆ (𝐹𝑦)))
103, 8, 9syl2anc 582 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → ((𝐹𝑥) ⊆ 𝑦𝑥 ⊆ (𝐹𝑦)))
1110anbi2d 628 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))
1211rexbidva 3166 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))
1312imbi2d 339 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) → (((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦)))))
1413ralbidv 3167 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦)))))
1514pm5.32da 577 . . 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 278 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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wrex 3059  wss 3944  ccnv 5677  dom cdm 5678  cima 5681  Fun wfun 6543  wf 6545  cfv 6549  (class class class)co 7419  TopOnctopon 22856   CnP ccnp 23173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-top 22840  df-topon 22857  df-cnp 23176
This theorem is referenced by:  cncnpi  23226  cnpdis  23241
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