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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 2141 | . . 3 | |
2 | 1 | cbvexv 1836 | . 2 |
3 | opelxp 4392 | . . . . . . . . . 10 | |
4 | fvres 5219 | . . . . . . . . . . . 12 | |
5 | vex 2604 | . . . . . . . . . . . . 13 | |
6 | vex 2604 | . . . . . . . . . . . . 13 | |
7 | 5, 6 | op1st 5793 | . . . . . . . . . . . 12 |
8 | 4, 7 | syl6req 2130 | . . . . . . . . . . 11 |
9 | f1stres 5806 | . . . . . . . . . . . . 13 | |
10 | ffn 5066 | . . . . . . . . . . . . 13 | |
11 | 9, 10 | ax-mp 7 | . . . . . . . . . . . 12 |
12 | fnfvelrn 5320 | . . . . . . . . . . . 12 | |
13 | 11, 12 | mpan 414 | . . . . . . . . . . 11 |
14 | 8, 13 | eqeltrd 2155 | . . . . . . . . . 10 |
15 | 3, 14 | sylbir 133 | . . . . . . . . 9 |
16 | 15 | expcom 114 | . . . . . . . 8 |
17 | 16 | exlimiv 1529 | . . . . . . 7 |
18 | 17 | ssrdv 3005 | . . . . . 6 |
19 | frn 5072 | . . . . . . 7 | |
20 | 9, 19 | ax-mp 7 | . . . . . 6 |
21 | 18, 20 | jctil 305 | . . . . 5 |
22 | eqss 3014 | . . . . 5 | |
23 | 21, 22 | sylibr 132 | . . . 4 |
24 | 23, 9 | jctil 305 | . . 3 |
25 | dffo2 5130 | . . 3 | |
26 | 24, 25 | sylibr 132 | . 2 |
27 | 2, 26 | sylbir 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 wss 2973 cop 3401 cxp 4361 crn 4364 cres 4365 wfn 4917 wf 4918 wfo 4920 cfv 4922 c1st 5785 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 df-1st 5787 |
This theorem is referenced by: 1stconst 5862 |
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