| 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 6583 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
| 2 | brrelex2 5739 | . . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V) | |
| 3 | 1, 2 | sylan 580 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V) |
| 4 | breq2 5147 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝐵)) | |
| 5 | 4 | anbi2d 630 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑦) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵))) |
| 6 | eqeq2 2749 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) = 𝑦 ↔ (𝐹‘𝐴) = 𝐵)) | |
| 7 | 5, 6 | imbi12d 344 | . . . 4 ⊢ (𝑦 = 𝐵 → (((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) ↔ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵))) |
| 8 | funeu 6591 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦) | |
| 9 | tz6.12-1 6929 | . . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | |
| 10 | 8, 9 | sylan2 593 | . . . . 5 ⊢ ((𝐴𝐹𝑦 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑦)) → (𝐹‘𝐴) = 𝑦) |
| 11 | 10 | anabss7 673 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) |
| 12 | 7, 11 | vtoclg 3554 | . . 3 ⊢ (𝐵 ∈ V → ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵)) |
| 13 | 3, 12 | mpcom 38 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵) |
| 14 | 13 | ex 412 | 1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!weu 2568 Vcvv 3480 class class class wbr 5143 Rel wrel 5690 Fun wfun 6555 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: funopfv 6958 fnbrfvb 6959 fvelima2 6961 fvelima 6974 fvelimad 6976 fvi 6985 opabiota 6991 fmptco 7149 fliftfun 7332 fliftval 7336 tfrlem5 8420 fpwwe2 10683 nqerid 10973 sum0 15757 sumz 15758 fsumsers 15764 isumclim 15793 ntrivcvgfvn0 15935 ntrivcvgtail 15936 zprodn0 15975 iprodclim 16034 idinv 17833 cnextfvval 24073 cnextfres 24077 dvadd 25977 dvmul 25978 dvco 25985 dvcj 25988 dvrec 25993 dvcnv 26015 dvef 26018 ftc1cn 26084 ulmdv 26446 minvecolem4b 30897 minvecolem4 30899 hlimuni 31257 chscllem4 31659 fmptcof2 32667 fvtransport 36033 fvray 36142 fvline 36145 ftc1cnnc 37699 iscard4 43546 frege124d 43774 |
| Copyright terms: Public domain | W3C validator |