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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaco 5152 |
. . . . 5
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2 | fnsnfv 5596 |
. . . . . 6
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3 | 2 | imaeq2d 4988 |
. . . . 5
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4 | 1, 3 | eqtr4id 2241 |
. . . 4
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5 | 4 | eleq2d 2259 |
. . 3
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6 | 5 | iotabidv 5218 |
. 2
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7 | dffv3g 5530 |
. . 3
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8 | 7 | adantl 277 |
. 2
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9 | funfvex 5551 |
. . . 4
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10 | 9 | funfni 5335 |
. . 3
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11 | dffv3g 5530 |
. . 3
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12 | 10, 11 | syl 14 |
. 2
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13 | 6, 8, 12 | 3eqtr4d 2232 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 |
This theorem is referenced by: fvco 5607 fvco3 5608 ofco 6125 updjudhcoinlf 7109 updjudhcoinrg 7110 updjud 7111 caseinl 7120 caseinr 7121 ctm 7138 enomnilem 7166 enmkvlem 7189 enwomnilem 7197 ringidvalg 13315 lidlvalg 13787 rspvalg 13788 |
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