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Mirrors > Home > MPE Home > Th. List > Mathboxes > funbrafv2b | Structured version Visualization version GIF version |
Description: Function value in terms of a binary relation, analogous to funbrfv2b 6759. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
funbrafv2b | ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 6386 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | releldm 5802 | . . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹) | |
3 | 2 | ex 416 | . . . 4 ⊢ (Rel 𝐹 → (𝐴𝐹𝐵 → 𝐴 ∈ dom 𝐹)) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → 𝐴 ∈ dom 𝐹)) |
5 | 4 | pm4.71rd 566 | . 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ 𝐴𝐹𝐵))) |
6 | funbrafvb 44274 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | |
7 | 6 | pm5.32da 582 | . 2 ⊢ (Fun 𝐹 → ((𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵) ↔ (𝐴 ∈ dom 𝐹 ∧ 𝐴𝐹𝐵))) |
8 | 5, 7 | bitr4d 285 | 1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 dom cdm 5540 Rel wrel 5545 Fun wfun 6363 '''cafv 44235 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 |
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 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-int 4850 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-res 5552 df-iota 6327 df-fun 6371 df-fn 6372 df-fv 6377 df-aiota 44203 df-dfat 44237 df-afv 44238 |
This theorem is referenced by: (None) |
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