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Mirrors > Home > MPE Home > Th. List > oprabco | Structured version Unicode version |
Description: Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.) |
Ref | Expression |
---|---|
oprabco.1 |
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oprabco.2 |
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oprabco.3 |
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Ref | Expression |
---|---|
oprabco |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oprabco.1 |
. . . 4
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2 | 1 | adantl 466 |
. . 3
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3 | oprabco.2 |
. . . 4
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4 | 3 | a1i 11 |
. . 3
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5 | dffn5 5839 |
. . . 4
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6 | 5 | biimpi 194 |
. . 3
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7 | fveq2 5792 |
. . 3
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8 | 2, 4, 6, 7 | fmpt2co 6759 |
. 2
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9 | oprabco.3 |
. 2
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10 | 8, 9 | syl6reqr 2511 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-id 4737 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-fv 5527 df-oprab 6197 df-mpt2 6198 df-1st 6680 df-2nd 6681 |
This theorem is referenced by: oprab2co 6761 |
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