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Mirrors > Home > ILE Home > Th. List > cnconst2 | Unicode version |
Description: A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
cnconst2 | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconst6g 5380 | . . 3 | |
2 | 1 | 3ad2ant3 1009 | . 2 TopOn TopOn |
3 | 2 | adantr 274 | . . . 4 TopOn TopOn |
4 | simpll3 1027 | . . . . . . . 8 TopOn TopOn | |
5 | simplr 520 | . . . . . . . 8 TopOn TopOn | |
6 | fvconst2g 5693 | . . . . . . . 8 | |
7 | 4, 5, 6 | syl2anc 409 | . . . . . . 7 TopOn TopOn |
8 | 7 | eleq1d 2233 | . . . . . 6 TopOn TopOn |
9 | simpll1 1025 | . . . . . . . . 9 TopOn TopOn TopOn | |
10 | toponmax 12570 | . . . . . . . . 9 TopOn | |
11 | 9, 10 | syl 14 | . . . . . . . 8 TopOn TopOn |
12 | simplr 520 | . . . . . . . 8 TopOn TopOn | |
13 | df-ima 4611 | . . . . . . . . 9 | |
14 | ssid 3157 | . . . . . . . . . . . . 13 | |
15 | xpssres 4913 | . . . . . . . . . . . . 13 | |
16 | 14, 15 | ax-mp 5 | . . . . . . . . . . . 12 |
17 | 16 | rneqi 4826 | . . . . . . . . . . 11 |
18 | rnxpss 5029 | . . . . . . . . . . 11 | |
19 | 17, 18 | eqsstri 3169 | . . . . . . . . . 10 |
20 | simprr 522 | . . . . . . . . . . 11 TopOn TopOn | |
21 | 20 | snssd 3712 | . . . . . . . . . 10 TopOn TopOn |
22 | 19, 21 | sstrid 3148 | . . . . . . . . 9 TopOn TopOn |
23 | 13, 22 | eqsstrid 3183 | . . . . . . . 8 TopOn TopOn |
24 | eleq2 2228 | . . . . . . . . . 10 | |
25 | imaeq2 4936 | . . . . . . . . . . 11 | |
26 | 25 | sseq1d 3166 | . . . . . . . . . 10 |
27 | 24, 26 | anbi12d 465 | . . . . . . . . 9 |
28 | 27 | rspcev 2825 | . . . . . . . 8 |
29 | 11, 12, 23, 28 | syl12anc 1225 | . . . . . . 7 TopOn TopOn |
30 | 29 | expr 373 | . . . . . 6 TopOn TopOn |
31 | 8, 30 | sylbid 149 | . . . . 5 TopOn TopOn |
32 | 31 | ralrimiva 2537 | . . . 4 TopOn TopOn |
33 | simpl1 989 | . . . . 5 TopOn TopOn TopOn | |
34 | simpl2 990 | . . . . 5 TopOn TopOn TopOn | |
35 | simpr 109 | . . . . 5 TopOn TopOn | |
36 | iscnp 12746 | . . . . 5 TopOn TopOn | |
37 | 33, 34, 35, 36 | syl3anc 1227 | . . . 4 TopOn TopOn |
38 | 3, 32, 37 | mpbir2and 933 | . . 3 TopOn TopOn |
39 | 38 | ralrimiva 2537 | . 2 TopOn TopOn |
40 | cncnp 12777 | . . 3 TopOn TopOn | |
41 | 40 | 3adant3 1006 | . 2 TopOn TopOn |
42 | 2, 39, 41 | mpbir2and 933 | 1 TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wral 2442 wrex 2443 wss 3111 csn 3570 cxp 4596 crn 4599 cres 4600 cima 4601 wf 5178 cfv 5182 (class class class)co 5836 TopOnctopon 12555 ccn 12732 ccnp 12733 |
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-cn 12735 df-cnp 12736 |
This theorem is referenced by: cnconst 12781 cnmptc 12829 |
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