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Theorem iscnp3 12382
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 12378 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
2 ffun 5275 . . . . . . . . . 10 (𝐹:𝑋𝑌 → Fun 𝐹)
32ad2antlr 480 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → Fun 𝐹)
4 toponss 12203 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
54adantlr 468 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → 𝑥𝑋)
6 fdm 5278 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
76ad2antlr 480 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → dom 𝐹 = 𝑋)
85, 7sseqtrrd 3136 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → 𝑥 ⊆ dom 𝐹)
9 funimass3 5536 . . . . . . . . 9 ((Fun 𝐹𝑥 ⊆ dom 𝐹) → ((𝐹𝑥) ⊆ 𝑦𝑥 ⊆ (𝐹𝑦)))
103, 8, 9syl2anc 408 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → ((𝐹𝑥) ⊆ 𝑦𝑥 ⊆ (𝐹𝑦)))
1110anbi2d 459 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐽) → ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))
1211rexbidva 2434 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))
1312imbi2d 229 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) → (((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦)))))
1413ralbidv 2437 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦)))))
1514pm5.32da 447 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))))
16153ad2ant1 1002 . 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 187 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 set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  wral 2416  wrex 2417  wss 3071  ccnv 4538  dom cdm 4539  cima 4542  Fun wfun 5117  wf 5119  cfv 5123  (class class class)co 5774  TopOnctopon 12187   CnP ccnp 12365
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-top 12175  df-topon 12188  df-cnp 12368
This theorem is referenced by:  cncnpi  12407  cnpdis  12421
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