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Mirrors > Home > ILE Home > Th. List > tpossym | Unicode version |
Description: Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
tpossym | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposfn 6138 | . . 3 tpos | |
2 | eqfnov2 5846 | . . 3 tpos tpos tpos | |
3 | 1, 2 | mpancom 418 | . 2 tpos tpos |
4 | eqcom 2119 | . . . 4 tpos tpos | |
5 | vex 2663 | . . . . . 6 | |
6 | vex 2663 | . . . . . 6 | |
7 | ovtposg 6124 | . . . . . 6 tpos | |
8 | 5, 6, 7 | mp2an 422 | . . . . 5 tpos |
9 | 8 | eqeq2i 2128 | . . . 4 tpos |
10 | 4, 9 | bitri 183 | . . 3 tpos |
11 | 10 | 2ralbii 2420 | . 2 tpos |
12 | 3, 11 | syl6bb 195 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1316 wcel 1465 wral 2393 cvv 2660 cxp 4507 wfn 5088 (class class class)co 5742 tpos ctpos 6109 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 df-ov 5745 df-tpos 6110 |
This theorem is referenced by: xmettpos 12466 |
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