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Mirrors > Home > ILE Home > Th. List > fvco2 | Unicode version |
Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.) |
Ref | Expression |
---|---|
fvco2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnsnfv 5253 | . . . . . 6 | |
2 | 1 | imaeq2d 4688 | . . . . 5 |
3 | imaco 4846 | . . . . 5 | |
4 | 2, 3 | syl6reqr 2132 | . . . 4 |
5 | 4 | eleq2d 2148 | . . 3 |
6 | 5 | iotabidv 4908 | . 2 |
7 | dffv3g 5194 | . . 3 | |
8 | 7 | adantl 271 | . 2 |
9 | funfvex 5212 | . . . 4 | |
10 | 9 | funfni 5019 | . . 3 |
11 | dffv3g 5194 | . . 3 | |
12 | 10, 11 | syl 14 | . 2 |
13 | 6, 8, 12 | 3eqtr4d 2123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cvv 2601 csn 3398 cima 4366 ccom 4367 cio 4885 wfn 4917 cfv 4922 |
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-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 |
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-v 2603 df-sbc 2816 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-br 3786 df-opab 3840 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-fv 4930 |
This theorem is referenced by: fvco 5264 fvco3 5265 ofco 5749 |
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