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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 2823 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2823 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | cnpf 21857 | . . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
4 | toponuni 21524 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
5 | 4 | feq2d 6502 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶𝑌)) |
6 | toponuni 21524 | . . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾) | |
7 | 6 | feq3d 6503 | . . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → (𝐹:∪ 𝐽⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾)) |
8 | 5, 7 | sylan9bb 512 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾)) |
9 | 3, 8 | syl5ibr 248 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶𝑌)) |
10 | 9 | 3impia 1113 | 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 398 ∧ w3a 1083 ∈ wcel 2114 ∪ cuni 4840 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 TopOnctopon 21520 CnP ccnp 21835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-top 21504 df-topon 21521 df-cnp 21838 |
This theorem is referenced by: iscnp4 21873 1stccnp 22072 txcnp 22230 ptcnplem 22231 ptcnp 22232 cnpflf2 22610 cnpflf 22611 flfcnp 22614 flfcnp2 22617 cnpfcf 22651 ghmcnp 22725 metcnpi3 23158 limcvallem 24471 cnplimc 24487 limccnp 24491 limccnp2 24492 ftc1lem3 24637 |
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