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Mirrors > Home > MPE Home > Th. List > nat1st2nd | Structured version Visualization version Unicode version |
Description: Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
natrcl.1 |
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nat1st2nd.2 |
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Ref | Expression |
---|---|
nat1st2nd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nat1st2nd.2 |
. 2
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2 | relfunc 15760 |
. . . 4
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3 | natrcl.1 |
. . . . . . 7
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4 | 3 | natrcl 15848 |
. . . . . 6
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5 | 1, 4 | syl 17 |
. . . . 5
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6 | 5 | simpld 461 |
. . . 4
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7 | 1st2nd 6836 |
. . . 4
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8 | 2, 6, 7 | sylancr 668 |
. . 3
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9 | 5 | simprd 465 |
. . . 4
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10 | 1st2nd 6836 |
. . . 4
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11 | 2, 9, 10 | sylancr 668 |
. . 3
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12 | 8, 11 | oveq12d 6306 |
. 2
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13 | 1, 12 | eleqtrd 2530 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-1st 6790 df-2nd 6791 df-ixp 7520 df-func 15756 df-nat 15841 |
This theorem is referenced by: fuccocl 15862 fuclid 15864 fucrid 15865 fucass 15866 fucsect 15870 invfuc 15872 fucpropd 15875 evlfcllem 16099 evlfcl 16100 curfuncf 16116 yonedalem3a 16152 yonedalem3b 16157 yonedainv 16159 yonffthlem 16160 |
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