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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 2733 | . . . . . . . 8 | |
2 | vex 2733 | . . . . . . . 8 | |
3 | 1, 2 | op1sta 5092 | . . . . . . 7 |
4 | 3 | eleq1i 2236 | . . . . . 6 |
5 | 4 | biimpri 132 | . . . . 5 |
6 | 5 | adantr 274 | . . . 4 |
7 | 6 | rgen2 2556 | . . 3 |
8 | sneq 3594 | . . . . . . 7 | |
9 | 8 | dmeqd 4813 | . . . . . 6 |
10 | 9 | unieqd 3807 | . . . . 5 |
11 | 10 | eleq1d 2239 | . . . 4 |
12 | 11 | ralxp 4754 | . . 3 |
13 | 7, 12 | mpbir 145 | . 2 |
14 | df-1st 6119 | . . . . 5 | |
15 | 14 | reseq1i 4887 | . . . 4 |
16 | ssv 3169 | . . . . 5 | |
17 | resmpt 4939 | . . . . 5 | |
18 | 16, 17 | ax-mp 5 | . . . 4 |
19 | 15, 18 | eqtri 2191 | . . 3 |
20 | 19 | fmpt 5646 | . 2 |
21 | 13, 20 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1348 wcel 2141 wral 2448 cvv 2730 wss 3121 csn 3583 cop 3586 cuni 3796 cmpt 4050 cxp 4609 cdm 4611 cres 4613 wf 5194 c1st 6117 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-1st 6119 |
This theorem is referenced by: fo1stresm 6140 1stcof 6142 tx1cn 13063 |
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