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Mirrors > Home > ILE Home > Th. List > fo1stresm | Unicode version |
Description: Onto mapping of a restriction of the (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
Ref | Expression |
---|---|
fo1stresm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2200 | . . 3 | |
2 | 1 | cbvexv 1890 | . 2 |
3 | opelxp 4564 | . . . . . . . . . 10 | |
4 | fvres 5438 | . . . . . . . . . . . 12 | |
5 | vex 2684 | . . . . . . . . . . . . 13 | |
6 | vex 2684 | . . . . . . . . . . . . 13 | |
7 | 5, 6 | op1st 6037 | . . . . . . . . . . . 12 |
8 | 4, 7 | syl6req 2187 | . . . . . . . . . . 11 |
9 | f1stres 6050 | . . . . . . . . . . . . 13 | |
10 | ffn 5267 | . . . . . . . . . . . . 13 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . . . 12 |
12 | fnfvelrn 5545 | . . . . . . . . . . . 12 | |
13 | 11, 12 | mpan 420 | . . . . . . . . . . 11 |
14 | 8, 13 | eqeltrd 2214 | . . . . . . . . . 10 |
15 | 3, 14 | sylbir 134 | . . . . . . . . 9 |
16 | 15 | expcom 115 | . . . . . . . 8 |
17 | 16 | exlimiv 1577 | . . . . . . 7 |
18 | 17 | ssrdv 3098 | . . . . . 6 |
19 | frn 5276 | . . . . . . 7 | |
20 | 9, 19 | ax-mp 5 | . . . . . 6 |
21 | 18, 20 | jctil 310 | . . . . 5 |
22 | eqss 3107 | . . . . 5 | |
23 | 21, 22 | sylibr 133 | . . . 4 |
24 | 23, 9 | jctil 310 | . . 3 |
25 | dffo2 5344 | . . 3 | |
26 | 24, 25 | sylibr 133 | . 2 |
27 | 2, 26 | sylbir 134 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 wss 3066 cop 3525 cxp 4532 crn 4535 cres 4536 wfn 5113 wf 5114 wfo 5116 cfv 5118 c1st 6029 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fo 5124 df-fv 5126 df-1st 6031 |
This theorem is referenced by: 1stconst 6111 |
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