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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnbrafv2b | Structured version Visualization version GIF version |
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6895. (Contributed by AV, 6-Sep-2022.) |
Ref | Expression |
---|---|
fnbrafv2b | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . 4 ⊢ (𝐹''''𝐵) = (𝐹''''𝐵) | |
2 | fundmdfat 45351 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝐹 defAt 𝐵) | |
3 | 2 | funfni 6608 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐹 defAt 𝐵) |
4 | dfatafv2ex 45435 | . . . . . 6 ⊢ (𝐹 defAt 𝐵 → (𝐹''''𝐵) ∈ V) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ V) |
6 | eqeq2 2748 | . . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐵) → ((𝐹''''𝐵) = 𝑥 ↔ (𝐹''''𝐵) = (𝐹''''𝐵))) | |
7 | breq2 5109 | . . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹''''𝐵))) | |
8 | 6, 7 | bibi12d 345 | . . . . . 6 ⊢ (𝑥 = (𝐹''''𝐵) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))) |
9 | 8 | adantl 482 | . . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 = (𝐹''''𝐵)) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))) |
10 | fneu 6612 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥) | |
11 | tz6.12c-afv2 45464 | . . . . . 6 ⊢ (∃!𝑥 𝐵𝐹𝑥 → ((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥)) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥)) |
13 | 5, 9, 12 | vtocld 3511 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))) |
14 | 1, 13 | mpbii 232 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹''''𝐵)) |
15 | breq2 5109 | . . 3 ⊢ ((𝐹''''𝐵) = 𝐶 → (𝐵𝐹(𝐹''''𝐵) ↔ 𝐵𝐹𝐶)) | |
16 | 14, 15 | syl5ibcom 244 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 → 𝐵𝐹𝐶)) |
17 | fnfun 6602 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
18 | funbrafv2 45469 | . . . 4 ⊢ (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶)) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶)) |
20 | 19 | adantr 481 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶)) |
21 | 16, 20 | impbid 211 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃!weu 2566 Vcvv 3445 class class class wbr 5105 Fun wfun 6490 Fn wfn 6491 defAt wdfat 45338 ''''cafv2 45430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-res 5645 df-iota 6448 df-fun 6498 df-fn 6499 df-dfat 45341 df-afv2 45431 |
This theorem is referenced by: fnopafv2b 45471 funbrafv22b 45472 |
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