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Mirrors > Home > ILE Home > Th. List > f1stres | Unicode version |
Description: Mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
f1stres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2715 | . . . . . . . 8 | |
2 | vex 2715 | . . . . . . . 8 | |
3 | 1, 2 | op1sta 5067 | . . . . . . 7 |
4 | 3 | eleq1i 2223 | . . . . . 6 |
5 | 4 | biimpri 132 | . . . . 5 |
6 | 5 | adantr 274 | . . . 4 |
7 | 6 | rgen2 2543 | . . 3 |
8 | sneq 3571 | . . . . . . 7 | |
9 | 8 | dmeqd 4788 | . . . . . 6 |
10 | 9 | unieqd 3783 | . . . . 5 |
11 | 10 | eleq1d 2226 | . . . 4 |
12 | 11 | ralxp 4729 | . . 3 |
13 | 7, 12 | mpbir 145 | . 2 |
14 | df-1st 6088 | . . . . 5 | |
15 | 14 | reseq1i 4862 | . . . 4 |
16 | ssv 3150 | . . . . 5 | |
17 | resmpt 4914 | . . . . 5 | |
18 | 16, 17 | ax-mp 5 | . . . 4 |
19 | 15, 18 | eqtri 2178 | . . 3 |
20 | 19 | fmpt 5617 | . 2 |
21 | 13, 20 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1335 wcel 2128 wral 2435 cvv 2712 wss 3102 csn 3560 cop 3563 cuni 3772 cmpt 4025 cxp 4584 cdm 4586 cres 4588 wf 5166 c1st 6086 |
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-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-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
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-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 df-1st 6088 |
This theorem is referenced by: fo1stresm 6109 1stcof 6111 tx1cn 12680 |
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