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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 6797 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
2 | df-br 5031 | . . 3 ⊢ (𝐴𝐹(𝐹‘𝐴) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
3 | 1, 2 | sylibr 237 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐴𝐹(𝐹‘𝐴)) |
4 | funrel 6341 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | releldm 5778 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹(𝐹‘𝐴)) → 𝐴 ∈ dom 𝐹) | |
6 | 4, 5 | sylan 583 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹(𝐹‘𝐴)) → 𝐴 ∈ dom 𝐹) |
7 | 3, 6 | impbida 800 | 1 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 dom cdm 5519 Rel wrel 5524 Fun wfun 6318 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: fmptco 6868 fpwwe2lem13 10053 fpwwe2 10054 climdm 14903 invco 17033 ffthiso 17191 fuciso 17237 setciso 17343 catciso 17359 lmcau 23917 dvcnp 24522 dvadd 24543 dvmul 24544 dvaddf 24545 dvmulf 24546 dvco 24550 dvcof 24551 dvcjbr 24552 dvcnvlem 24579 dvferm1 24588 dvferm2 24590 ulmdm 24988 ulmdvlem3 24997 minvecolem4a 28660 hlimf 29020 hhsscms 29061 occllem 29086 occl 29087 chscllem4 29423 fmptcof2 30420 heiborlem9 35257 bfplem1 35260 iscard4 40241 xlimdm 42499 rngciso 44606 rngcisoALTV 44618 ringciso 44657 ringcisoALTV 44681 |
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