Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6812 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
2 | df-br 5058 | . . 3 ⊢ (𝐴𝐹(𝐹‘𝐴) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
3 | 1, 2 | sylibr 235 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐴𝐹(𝐹‘𝐴)) |
4 | funrel 6365 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | releldm 5807 | . . 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 797 | 1 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 〈cop 4563 class class class wbr 5057 dom cdm 5548 Rel wrel 5553 Fun wfun 6342 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: fmptco 6883 fpwwe2lem13 10052 fpwwe2 10053 climdm 14899 invco 17029 ffthiso 17187 fuciso 17233 setciso 17339 catciso 17355 lmcau 23843 dvcnp 24443 dvadd 24464 dvmul 24465 dvaddf 24466 dvmulf 24467 dvco 24471 dvcof 24472 dvcjbr 24473 dvcnvlem 24500 dvferm1 24509 dvferm2 24511 ulmdm 24908 ulmdvlem3 24917 minvecolem4a 28581 hlimf 28941 hhsscms 28982 occllem 29007 occl 29008 chscllem4 29344 fmptcof2 30330 heiborlem9 34978 bfplem1 34981 iscard4 39778 xlimdm 42014 rngciso 44181 rngcisoALTV 44193 ringciso 44232 ringcisoALTV 44256 |
Copyright terms: Public domain | W3C validator |