Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > funbrfv | Structured version Visualization version GIF version |
Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
funbrfv | ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 6375 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | brrelex2 5588 | . . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V) | |
3 | 1, 2 | sylan 583 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V) |
4 | breq2 5043 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝐵)) | |
5 | 4 | anbi2d 632 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑦) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵))) |
6 | eqeq2 2748 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) = 𝑦 ↔ (𝐹‘𝐴) = 𝐵)) | |
7 | 5, 6 | imbi12d 348 | . . . 4 ⊢ (𝑦 = 𝐵 → (((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) ↔ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵))) |
8 | funeu 6383 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦) | |
9 | tz6.12-1 6717 | . . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | |
10 | 8, 9 | sylan2 596 | . . . . 5 ⊢ ((𝐴𝐹𝑦 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑦)) → (𝐹‘𝐴) = 𝑦) |
11 | 10 | anabss7 673 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) |
12 | 7, 11 | vtoclg 3471 | . . 3 ⊢ (𝐵 ∈ V → ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵)) |
13 | 3, 12 | mpcom 38 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵) |
14 | 13 | ex 416 | 1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃!weu 2567 Vcvv 3398 class class class wbr 5039 Rel wrel 5541 Fun wfun 6352 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 |
This theorem is referenced by: funopfv 6742 fnbrfvb 6743 fvelima 6756 fvelimad 6757 fvi 6765 opabiota 6772 fmptco 6922 fliftfun 7099 fliftval 7103 tfrlem5 8094 fpwwe2 10222 nqerid 10512 sum0 15250 sumz 15251 fsumsers 15257 isumclim 15284 ntrivcvgfvn0 15426 ntrivcvgtail 15427 zprodn0 15464 iprodclim 15523 idinv 17248 cnextfvval 22916 cnextfres 22920 dvadd 24791 dvmul 24792 dvco 24798 dvcj 24801 dvrec 24806 dvcnv 24828 dvef 24831 ftc1cn 24894 ulmdv 25249 minvecolem4b 28913 minvecolem4 28915 hlimuni 29273 chscllem4 29675 fmptcof2 30668 fvtransport 34020 fvray 34129 fvline 34132 ftc1cnnc 35535 iscard4 40766 frege124d 40987 fvelima2 42419 |
Copyright terms: Public domain | W3C validator |