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 6718. (Contributed by AV, 6-Sep-2022.) |
Ref | Expression |
---|---|
fnbrafv2b | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (𝐹''''𝐵) = (𝐹''''𝐵) | |
2 | fundmdfat 43377 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝐹 defAt 𝐵) | |
3 | 2 | funfni 6457 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐹 defAt 𝐵) |
4 | dfatafv2ex 43461 | . . . . . 6 ⊢ (𝐹 defAt 𝐵 → (𝐹''''𝐵) ∈ V) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ V) |
6 | eqeq2 2833 | . . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐵) → ((𝐹''''𝐵) = 𝑥 ↔ (𝐹''''𝐵) = (𝐹''''𝐵))) | |
7 | breq2 5070 | . . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹''''𝐵))) | |
8 | 6, 7 | bibi12d 348 | . . . . . 6 ⊢ (𝑥 = (𝐹''''𝐵) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))) |
9 | 8 | adantl 484 | . . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 = (𝐹''''𝐵)) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))) |
10 | fneu 6461 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥) | |
11 | tz6.12c-afv2 43490 | . . . . . 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 3556 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))) |
14 | 1, 13 | mpbii 235 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹''''𝐵)) |
15 | breq2 5070 | . . 3 ⊢ ((𝐹''''𝐵) = 𝐶 → (𝐵𝐹(𝐹''''𝐵) ↔ 𝐵𝐹𝐶)) | |
16 | 14, 15 | syl5ibcom 247 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 → 𝐵𝐹𝐶)) |
17 | fnfun 6453 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
18 | funbrafv2 43495 | . . . 4 ⊢ (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶)) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶)) |
20 | 19 | adantr 483 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶)) |
21 | 16, 20 | impbid 214 | 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 1537 ∈ wcel 2114 ∃!weu 2653 Vcvv 3494 class class class wbr 5066 Fun wfun 6349 Fn wfn 6350 defAt wdfat 43364 ''''cafv2 43456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fn 6358 df-dfat 43367 df-afv2 43457 |
This theorem is referenced by: fnopafv2b 43497 funbrafv22b 43498 |
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