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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 2622 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2622 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | cnpf 21051 | . . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
4 | toponuni 20719 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
5 | 4 | feq2d 6031 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶𝑌)) |
6 | toponuni 20719 | . . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾) | |
7 | 6 | feq3d 6032 | . . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → (𝐹:∪ 𝐽⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾)) |
8 | 5, 7 | sylan9bb 736 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾)) |
9 | 3, 8 | syl5ibr 236 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶𝑌)) |
10 | 9 | 3impia 1261 | 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 384 ∧ w3a 1037 ∈ wcel 1990 ∪ cuni 4436 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 TopOnctopon 20715 CnP ccnp 21029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-top 20699 df-topon 20716 df-cnp 21032 |
This theorem is referenced by: iscnp4 21067 1stccnp 21265 txcnp 21423 ptcnplem 21424 ptcnp 21425 cnpflf2 21804 cnpflf 21805 flfcnp 21808 flfcnp2 21811 cnpfcf 21845 ghmcnp 21918 metcnpi3 22351 limcvallem 23635 cnplimc 23651 limccnp 23655 limccnp2 23656 ftc1lem3 23801 |
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