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Mirrors > Home > MPE Home > Th. List > funfvbrb | Structured version Visualization version GIF version |
Description: Two ways to say that 𝐴 is in the domain of 𝐹. (Contributed by Mario Carneiro, 1-May-2014.) |
Ref | Expression |
---|---|
funfvbrb | ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvop 7051 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
2 | df-br 5149 | . . 3 ⊢ (𝐴𝐹(𝐹‘𝐴) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
3 | 1, 2 | sylibr 233 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐴𝐹(𝐹‘𝐴)) |
4 | funrel 6565 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | releldm 5943 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹(𝐹‘𝐴)) → 𝐴 ∈ dom 𝐹) | |
6 | 4, 5 | sylan 579 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹(𝐹‘𝐴)) → 𝐴 ∈ dom 𝐹) |
7 | 3, 6 | impbida 798 | 1 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2105 〈cop 4634 class class class wbr 5148 dom cdm 5676 Rel wrel 5681 Fun wfun 6537 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 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-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 |
This theorem is referenced by: fmptco 7129 fpwwe2lem12 10643 fpwwe2 10644 climdm 15505 invco 17725 ffthiso 17889 fuciso 17938 setciso 18051 catciso 18071 rngciso 20530 ringciso 20564 lmcau 25160 dvcnp 25767 dvadd 25790 dvmul 25791 dvaddf 25792 dvmulf 25793 dvco 25798 dvcof 25799 dvcjbr 25800 dvcnvlem 25827 dvferm1 25836 dvferm2 25838 ulmdm 26243 ulmdvlem3 26252 minvecolem4a 30562 hlimf 30922 hhsscms 30963 occllem 30988 occl 30989 chscllem4 31325 fmptcof2 32314 heiborlem9 37150 bfplem1 37153 iscard4 42746 xlimdm 45031 rngcisoALTV 47113 ringcisoALTV 47147 |
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