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Mirrors > Home > ILE Home > Th. List > 2ndinr | Unicode version |
Description: The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
2ndinr | inr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inr 6901 | . . . . 5 inr | |
2 | 1 | a1i 9 | . . . 4 inr |
3 | opeq2 3676 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | elex 2671 | . . . 4 | |
6 | 1on 6288 | . . . . 5 | |
7 | opexg 4120 | . . . . 5 | |
8 | 6, 7 | mpan 420 | . . . 4 |
9 | 2, 4, 5, 8 | fvmptd 5470 | . . 3 inr |
10 | 9 | fveq2d 5393 | . 2 inr |
11 | op2ndg 6017 | . . 3 | |
12 | 6, 11 | mpan 420 | . 2 |
13 | 10, 12 | eqtrd 2150 | 1 inr |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 wcel 1465 cvv 2660 cop 3500 cmpt 3959 con0 4255 cfv 5093 c2nd 6005 c1o 6274 inrcinr 6899 |
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-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 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-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fv 5101 df-2nd 6007 df-1o 6281 df-inr 6901 |
This theorem is referenced by: updjudhcoinrg 6934 |
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