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Mirrors > Home > MPE Home > Th. List > cnpf2 | Structured version Visualization version GIF version |
Description: A continuous function at point 𝑃 is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnpf2 | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2735 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | cnpf 23271 | . . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
4 | toponuni 22936 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
5 | 4 | feq2d 6723 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶𝑌)) |
6 | toponuni 22936 | . . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾) | |
7 | 6 | feq3d 6724 | . . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → (𝐹:∪ 𝐽⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾)) |
8 | 5, 7 | sylan9bb 509 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾)) |
9 | 3, 8 | imbitrrid 246 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶𝑌)) |
10 | 9 | 3impia 1116 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ∪ cuni 4912 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 TopOnctopon 22932 CnP ccnp 23249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-top 22916 df-topon 22933 df-cnp 23252 |
This theorem is referenced by: iscnp4 23287 1stccnp 23486 txcnp 23644 ptcnplem 23645 ptcnp 23646 cnpflf2 24024 cnpflf 24025 flfcnp 24028 flfcnp2 24031 cnpfcf 24065 ghmcnp 24139 metcnpi3 24575 limcvallem 25921 cnplimc 25937 limccnp 25941 limccnp2 25942 ftc1lem3 26094 |
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