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Mirrors > Home > ILE Home > Th. List > 1stinr | Unicode version |
Description: The first component of the value of a right injection is . (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
1stinr | inr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inr 6993 | . . . . 5 inr | |
2 | 1 | a1i 9 | . . . 4 inr |
3 | opeq2 3743 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | elex 2723 | . . . 4 | |
6 | 1on 6371 | . . . . 5 | |
7 | opexg 4189 | . . . . 5 | |
8 | 6, 7 | mpan 421 | . . . 4 |
9 | 2, 4, 5, 8 | fvmptd 5550 | . . 3 inr |
10 | 9 | fveq2d 5473 | . 2 inr |
11 | op1stg 6099 | . . 3 | |
12 | 6, 11 | mpan 421 | . 2 |
13 | 10, 12 | eqtrd 2190 | 1 inr |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 wcel 2128 cvv 2712 cop 3563 cmpt 4026 con0 4324 cfv 5171 c1st 6087 c1o 6357 inrcinr 6991 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-iota 5136 df-fun 5173 df-fv 5179 df-1st 6089 df-1o 6364 df-inr 6993 |
This theorem is referenced by: djune 7023 updjudhcoinrg 7026 |
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