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Mirrors > Home > MPE Home > Th. List > funbrfvb | Structured version Visualization version GIF version |
Description: Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.) |
Ref | Expression |
---|---|
funbrfvb | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfn 6578 | . 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | fnbrfvb 6944 | . 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | |
3 | 1, 2 | sylanb 580 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 dom cdm 5676 Fun wfun 6537 Fn wfn 6538 ‘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: funbrfv2b 6949 dfimafn 6954 funimass4 6956 eqfunresadj 7360 dcomex 10448 dvidlem 25764 taylthlem1 26224 dfimafnf 32294 funcnvmpt 32326 cantnf2 42540 frege124d 42977 frege129d 42979 ntrclsfv1 43271 ntrneifv1 43295 ntrneifv2 43296 |
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