Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnbrafvb | Structured version Visualization version GIF version |
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6712. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
fnbrafvb | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndm 6449 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
2 | eleq2 2901 | . . . . . . . 8 ⊢ (𝐴 = dom 𝐹 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹)) | |
3 | 2 | eqcoms 2829 | . . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹)) |
4 | 3 | biimpd 231 | . . . . . 6 ⊢ (dom 𝐹 = 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹)) |
6 | 5 | imp 409 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹) |
7 | snssi 4734 | . . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
8 | 7 | adantl 484 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴) |
9 | fnssresb 6463 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴)) | |
10 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴)) |
11 | 8, 10 | mpbird 259 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) Fn {𝐵}) |
12 | fnfun 6447 | . . . . 5 ⊢ ((𝐹 ↾ {𝐵}) Fn {𝐵} → Fun (𝐹 ↾ {𝐵})) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → Fun (𝐹 ↾ {𝐵})) |
14 | df-dfat 43312 | . . . . 5 ⊢ (𝐹 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵}))) | |
15 | afvfundmfveq 43331 | . . . . 5 ⊢ (𝐹 defAt 𝐵 → (𝐹'''𝐵) = (𝐹‘𝐵)) | |
16 | 14, 15 | sylbir 237 | . . . 4 ⊢ ((𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})) → (𝐹'''𝐵) = (𝐹‘𝐵)) |
17 | 6, 13, 16 | syl2anc 586 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) = (𝐹‘𝐵)) |
18 | 17 | eqeq1d 2823 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶)) |
19 | fnbrfvb 6712 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) | |
20 | 18, 19 | bitrd 281 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 {csn 4560 class class class wbr 5058 dom cdm 5549 ↾ cres 5551 Fun wfun 6343 Fn wfn 6344 ‘cfv 6349 defAt wdfat 43309 '''cafv 43310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 df-aiota 43279 df-dfat 43312 df-afv 43313 |
This theorem is referenced by: fnopafvb 43348 funbrafvb 43349 dfafn5a 43353 |
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