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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 6938. (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 6646 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
2 | eleq2 2816 | . . . . . . . 8 ⊢ (𝐴 = dom 𝐹 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹)) | |
3 | 2 | eqcoms 2734 | . . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹)) |
4 | 3 | biimpd 228 | . . . . . 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 406 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹) |
7 | snssi 4806 | . . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
8 | 7 | adantl 481 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴) |
9 | fnssresb 6666 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴)) | |
10 | 9 | adantr 480 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴)) |
11 | 8, 10 | mpbird 257 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) Fn {𝐵}) |
12 | fnfun 6643 | . . . . 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 46396 | . . . . 5 ⊢ (𝐹 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵}))) | |
15 | afvfundmfveq 46415 | . . . . 5 ⊢ (𝐹 defAt 𝐵 → (𝐹'''𝐵) = (𝐹‘𝐵)) | |
16 | 14, 15 | sylbir 234 | . . . 4 ⊢ ((𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})) → (𝐹'''𝐵) = (𝐹‘𝐵)) |
17 | 6, 13, 16 | syl2anc 583 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) = (𝐹‘𝐵)) |
18 | 17 | eqeq1d 2728 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶)) |
19 | fnbrfvb 6938 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) | |
20 | 18, 19 | bitrd 279 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 {csn 4623 class class class wbr 5141 dom cdm 5669 ↾ cres 5671 Fun wfun 6531 Fn wfn 6532 ‘cfv 6537 defAt wdfat 46393 '''cafv 46394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6489 df-fun 6539 df-fn 6540 df-fv 6545 df-aiota 46362 df-dfat 46396 df-afv 46397 |
This theorem is referenced by: fnopafvb 46432 funbrafvb 46433 dfafn5a 46437 |
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