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Mirrors > Home > ILE Home > Th. List > cnmpt2nd | Unicode version |
Description: The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmpt21.j | TopOn |
cnmpt21.k | TopOn |
Ref | Expression |
---|---|
cnmpt2nd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo2nd 6118 | . . . . . 6 | |
2 | fofn 5406 | . . . . . 6 | |
3 | 1, 2 | ax-mp 5 | . . . . 5 |
4 | ssv 3159 | . . . . 5 | |
5 | fnssres 5295 | . . . . 5 | |
6 | 3, 4, 5 | mp2an 423 | . . . 4 |
7 | dffn5im 5526 | . . . 4 | |
8 | 6, 7 | ax-mp 5 | . . 3 |
9 | fvres 5504 | . . . 4 | |
10 | 9 | mpteq2ia 4062 | . . 3 |
11 | vex 2724 | . . . . 5 | |
12 | vex 2724 | . . . . 5 | |
13 | 11, 12 | op2ndd 6109 | . . . 4 |
14 | 13 | mpompt 5925 | . . 3 |
15 | 8, 10, 14 | 3eqtri 2189 | . 2 |
16 | cnmpt21.j | . . 3 TopOn | |
17 | cnmpt21.k | . . 3 TopOn | |
18 | tx2cn 12817 | . . 3 TopOn TopOn | |
19 | 16, 17, 18 | syl2anc 409 | . 2 |
20 | 15, 19 | eqeltrrid 2252 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1342 wcel 2135 cvv 2721 wss 3111 cmpt 4037 cxp 4596 cres 4600 wfn 5177 wfo 5180 cfv 5182 (class class class)co 5836 cmpo 5838 c2nd 6099 TopOnctopon 12555 ccn 12732 ctx 12799 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-topgen 12519 df-top 12543 df-topon 12556 df-bases 12588 df-cn 12735 df-tx 12800 |
This theorem is referenced by: cnmptcom 12845 txhmeo 12866 txswaphmeo 12868 divcnap 13102 cnrehmeocntop 13140 |
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