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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 6237 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
2 | df-br 4584 | . . 3 ⊢ (𝐴𝐹(𝐹‘𝐴) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
3 | 1, 2 | sylibr 223 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐴𝐹(𝐹‘𝐴)) |
4 | funrel 5821 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | releldm 5279 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹(𝐹‘𝐴)) → 𝐴 ∈ dom 𝐹) | |
6 | 4, 5 | sylan 487 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹(𝐹‘𝐴)) → 𝐴 ∈ dom 𝐹) |
7 | 3, 6 | impbida 873 | 1 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 dom cdm 5038 Rel wrel 5043 Fun wfun 5798 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: fmptco 6303 fpwwe2lem13 9343 fpwwe2 9344 climdm 14133 invco 16254 funciso 16357 ffthiso 16412 fuciso 16458 setciso 16564 catciso 16580 lmcau 22919 dvcnp 23488 dvadd 23509 dvmul 23510 dvaddf 23511 dvmulf 23512 dvco 23516 dvcof 23517 dvcjbr 23518 dvcnvlem 23543 dvferm1 23552 dvferm2 23554 ulmdm 23951 ulmdvlem3 23960 minvecolem4a 27117 hlimf 27478 hhsscms 27520 occllem 27546 occl 27547 chscllem4 27883 fmptcof2 28839 heiborlem9 32788 bfplem1 32791 rngciso 41774 rngcisoALTV 41786 ringciso 41825 ringcisoALTV 41849 |
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