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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 6586 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | funmpt2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funeqi 6569 | . 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)) |
4 | 1, 3 | mpbir 230 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ↦ cmpt 5231 Fun wfun 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-fun 6545 |
This theorem is referenced by: pwfilem 9176 cantnfp1lem1 9672 tz9.12lem2 9782 tz9.12lem3 9783 rankf 9788 djuun 9920 cardf2 9937 fin23lem30 10336 hashf1rn 14311 oppccatf 17673 funtopon 22421 qustgpopn 23623 ustn0 23724 cphsscph 24767 ipasslem8 30085 xppreima2 31871 funcnvmpt 31887 mptiffisupp 31910 fsuppcurry1 31945 fsuppcurry2 31946 gsummpt2co 32195 zarclsint 32847 zartopn 32850 zarmxt1 32855 zarcmplem 32856 brsiga 33176 sseqval 33382 ballotlem7 33529 sinccvglem 34652 bj-evalfun 35949 bj-ccinftydisj 36089 bj-elccinfty 36090 bj-minftyccb 36101 iscard4 42274 harval3 42279 comptiunov2i 42447 icccncfext 44593 stoweidlem27 44733 stirlinglem14 44793 fourierdlem70 44882 fourierdlem71 44883 hoi2toco 45313 mptcfsupp 47046 lcoc0 47093 lincresunit2 47149 |
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