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Mirrors > Home > ILE Home > Th. List > fmptcos | Unicode version |
Description: Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fmptcof.1 |
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fmptcof.2 |
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fmptcof.3 |
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Ref | Expression |
---|---|
fmptcos |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptcof.1 |
. 2
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2 | fmptcof.2 |
. 2
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3 | fmptcof.3 |
. . 3
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4 | nfcv 2235 |
. . . 4
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5 | nfcsb1v 2977 |
. . . 4
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6 | csbeq1a 2955 |
. . . 4
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7 | 4, 5, 6 | cbvmpt 3955 |
. . 3
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8 | 3, 7 | syl6eq 2143 |
. 2
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9 | csbeq1 2950 |
. 2
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10 | 1, 2, 8, 9 | fmptcof 5504 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 |
This theorem is referenced by: fmpt2co 6019 |
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