Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funbrafv | Structured version Visualization version GIF version |
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6709. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
funbrafv | ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 6365 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | releldm 5807 | . . . . . . . 8 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹) | |
3 | funbrafvb 43346 | . . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | |
4 | 3 | biimprd 250 | . . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)) |
5 | 4 | expcom 416 | . . . . . . . 8 ⊢ (𝐴 ∈ dom 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))) |
6 | 2, 5 | syl 17 | . . . . . . 7 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))) |
7 | 6 | ex 415 | . . . . . 6 ⊢ (Rel 𝐹 → (𝐴𝐹𝐵 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))) |
8 | 7 | com14 96 | . . . . 5 ⊢ (𝐴𝐹𝐵 → (𝐴𝐹𝐵 → (Fun 𝐹 → (Rel 𝐹 → (𝐹'''𝐴) = 𝐵)))) |
9 | 8 | pm2.43i 52 | . . . 4 ⊢ (𝐴𝐹𝐵 → (Fun 𝐹 → (Rel 𝐹 → (𝐹'''𝐴) = 𝐵))) |
10 | 9 | com13 88 | . . 3 ⊢ (Rel 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))) |
11 | 1, 10 | syl 17 | . 2 ⊢ (Fun 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))) |
12 | 11 | pm2.43i 52 | 1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 dom cdm 5548 Rel wrel 5553 Fun wfun 6342 '''cafv 43307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-int 4868 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-res 5560 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-aiota 43276 df-dfat 43309 df-afv 43310 |
This theorem is referenced by: afvelima 43357 |
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