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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 580 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹(𝐹‘𝐴)) → 𝐴 ∈ dom 𝐹) |
7 | 3, 6 | impbida 799 | 1 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ⟨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 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 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-ne 2941 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-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 10639 fpwwe2 10640 climdm 15502 invco 17722 ffthiso 17884 fuciso 17932 setciso 18045 catciso 18065 lmcau 25054 dvcnp 25660 dvadd 25681 dvmul 25682 dvaddf 25683 dvmulf 25684 dvco 25688 dvcof 25689 dvcjbr 25690 dvcnvlem 25717 dvferm1 25726 dvferm2 25728 ulmdm 26129 ulmdvlem3 26138 minvecolem4a 30385 hlimf 30745 hhsscms 30786 occllem 30811 occl 30812 chscllem4 31148 fmptcof2 32137 heiborlem9 36990 bfplem1 36993 iscard4 42586 xlimdm 44872 rngciso 46969 rngcisoALTV 46981 ringciso 47020 ringcisoALTV 47044 |
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