![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > funressnbrafv2 | Structured version Visualization version GIF version |
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6942. (Contributed by AV, 7-Sep-2022.) |
Ref | Expression |
---|---|
funressnbrafv2 | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpllr 775 | . . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → 𝐵 ∈ 𝑊) | |
2 | eleq1 2816 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊)) | |
3 | 2 | anbi2d 628 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))) |
4 | 3 | anbi1d 629 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})))) |
5 | breq2 5146 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝐵)) | |
6 | 4, 5 | anbi12d 630 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) ↔ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵))) |
7 | eqeq2 2739 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵)) | |
8 | 6, 7 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝐵 → (((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) ↔ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))) |
9 | id 22 | . . . . 5 ⊢ (𝐴𝐹𝑥 → 𝐴𝐹𝑥) | |
10 | funressneu 46352 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥) | |
11 | 10 | 3expa 1116 | . . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥) |
12 | tz6.12-1-afv2 46544 | . . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) | |
13 | 9, 11, 12 | syl2an2 685 | . . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) |
14 | 8, 13 | vtoclg 3538 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)) |
15 | 1, 14 | mpcom 38 | . 2 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵) |
16 | 15 | ex 412 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃!weu 2557 {csn 4624 class class class wbr 5142 ↾ cres 5674 Fun wfun 6536 ''''cafv2 46511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-res 5684 df-iota 6494 df-fun 6544 df-fn 6545 df-dfat 46422 df-afv2 46512 |
This theorem is referenced by: dfatbrafv2b 46548 |
Copyright terms: Public domain | W3C validator |