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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 2220 | . . 3 | |
2 | 1 | cbvexv 1898 | . 2 |
3 | opelxp 4616 | . . . . . . . . . 10 | |
4 | fvres 5492 | . . . . . . . . . . . 12 | |
5 | vex 2715 | . . . . . . . . . . . . 13 | |
6 | vex 2715 | . . . . . . . . . . . . 13 | |
7 | 5, 6 | op1st 6094 | . . . . . . . . . . . 12 |
8 | 4, 7 | eqtr2di 2207 | . . . . . . . . . . 11 |
9 | f1stres 6107 | . . . . . . . . . . . . 13 | |
10 | ffn 5319 | . . . . . . . . . . . . 13 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . . . 12 |
12 | fnfvelrn 5599 | . . . . . . . . . . . 12 | |
13 | 11, 12 | mpan 421 | . . . . . . . . . . 11 |
14 | 8, 13 | eqeltrd 2234 | . . . . . . . . . 10 |
15 | 3, 14 | sylbir 134 | . . . . . . . . 9 |
16 | 15 | expcom 115 | . . . . . . . 8 |
17 | 16 | exlimiv 1578 | . . . . . . 7 |
18 | 17 | ssrdv 3134 | . . . . . 6 |
19 | frn 5328 | . . . . . . 7 | |
20 | 9, 19 | ax-mp 5 | . . . . . 6 |
21 | 18, 20 | jctil 310 | . . . . 5 |
22 | eqss 3143 | . . . . 5 | |
23 | 21, 22 | sylibr 133 | . . . 4 |
24 | 23, 9 | jctil 310 | . . 3 |
25 | dffo2 5396 | . . 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 1335 wex 1472 wcel 2128 wss 3102 cop 3563 cxp 4584 crn 4587 cres 4588 wfn 5165 wf 5166 wfo 5168 cfv 5170 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-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
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-fo 5176 df-fv 5178 df-1st 6088 |
This theorem is referenced by: 1stconst 6168 |
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