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Mirrors > Home > MPE Home > Th. List > funmpt2 | Structured version Visualization version GIF version |
Description: Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) |
Ref | Expression |
---|---|
funmpt2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
funmpt2 | ⊢ Fun 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 6587 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | funmpt2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funeqi 6570 | . 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)) |
4 | 1, 3 | mpbir 230 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ↦ cmpt 5232 Fun wfun 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-fun 6546 |
This theorem is referenced by: pwfilem 9177 cantnfp1lem1 9673 tz9.12lem2 9783 tz9.12lem3 9784 rankf 9789 djuun 9921 cardf2 9938 fin23lem30 10337 hashf1rn 14312 oppccatf 17674 funtopon 22422 qustgpopn 23624 ustn0 23725 cphsscph 24768 ipasslem8 30090 xppreima2 31876 funcnvmpt 31892 mptiffisupp 31915 fsuppcurry1 31950 fsuppcurry2 31951 gsummpt2co 32200 zarclsint 32852 zartopn 32855 zarmxt1 32860 zarcmplem 32861 brsiga 33181 sseqval 33387 ballotlem7 33534 sinccvglem 34657 bj-evalfun 35954 bj-ccinftydisj 36094 bj-elccinfty 36095 bj-minftyccb 36106 iscard4 42284 harval3 42289 comptiunov2i 42457 icccncfext 44603 stoweidlem27 44743 stirlinglem14 44803 fourierdlem70 44892 fourierdlem71 44893 hoi2toco 45323 mptcfsupp 47056 lcoc0 47103 lincresunit2 47159 |
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