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Mirrors > Home > ILE Home > Th. List > txval | Unicode version |
Description: Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
txval.1 |
Ref | Expression |
---|---|
txval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2732 | . . . 4 | |
2 | 1 | adantr 274 | . . 3 |
3 | elex 2732 | . . . 4 | |
4 | 3 | adantl 275 | . . 3 |
5 | mpoexga 6172 | . . . 4 | |
6 | rnexg 4863 | . . . 4 | |
7 | tgvalex 12597 | . . . 4 | |
8 | 5, 6, 7 | 3syl 17 | . . 3 |
9 | mpoeq12 5893 | . . . . . 6 | |
10 | 9 | rneqd 4827 | . . . . 5 |
11 | 10 | fveq2d 5484 | . . . 4 |
12 | df-tx 12800 | . . . 4 | |
13 | 11, 12 | ovmpoga 5962 | . . 3 |
14 | 2, 4, 8, 13 | syl3anc 1227 | . 2 |
15 | txval.1 | . . 3 | |
16 | 15 | fveq2i 5483 | . 2 |
17 | 14, 16 | eqtr4di 2215 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cvv 2721 cxp 4596 crn 4599 cfv 5182 (class class class)co 5836 cmpo 5838 ctg 12513 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-topgen 12519 df-tx 12800 |
This theorem is referenced by: eltx 12806 txtop 12807 txtopon 12809 txopn 12812 txss12 12813 txbasval 12814 txcnp 12818 txcnmpt 12820 txrest 12823 txlm 12826 xmettxlem 13056 xmettx 13057 |
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