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| Description: Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) | 
| Ref | Expression | 
|---|---|
| dfdfat2 | ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-dfat 47136 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
| 2 | relres 6022 | . . . 4 ⊢ Rel (𝐹 ↾ {𝐴}) | |
| 3 | dffun8 6593 | . . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)) | |
| 4 | 2, 3 | mpbiran 709 | . . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) | 
| 5 | 4 | anbi2i 623 | . 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)) | 
| 6 | brres 6003 | . . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))) | |
| 7 | 6 | elv 3484 | . . . . . . 7 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)) | 
| 8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))) | 
| 9 | 8 | eubidv 2585 | . . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → (∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))) | 
| 10 | 9 | ralbidv 3177 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))) | 
| 11 | eldmressnsn 6041 | . . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴})) | |
| 12 | eldmressn 47054 | . . . . 5 ⊢ (𝑥 ∈ dom (𝐹 ↾ {𝐴}) → 𝑥 = 𝐴) | |
| 13 | velsn 4641 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
| 14 | 13 | biimpri 228 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑥 ∈ {𝐴}) | 
| 15 | breq1 5145 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
| 16 | 15 | anbi2d 630 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ {𝐴} ∧ 𝐴𝐹𝑦))) | 
| 17 | 14, 16 | mpbirand 707 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ 𝐴𝐹𝑦)) | 
| 18 | 17 | eubidv 2585 | . . . . 5 ⊢ (𝑥 = 𝐴 → (∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦)) | 
| 19 | 11, 12, 18 | ralbinrald 47139 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦)) | 
| 20 | 10, 19 | bitrd 279 | . . 3 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦)) | 
| 21 | 20 | pm5.32i 574 | . 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦)) | 
| 22 | 1, 5, 21 | 3bitri 297 | 1 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃!weu 2567 ∀wral 3060 Vcvv 3479 {csn 4625 class class class wbr 5142 dom cdm 5684 ↾ cres 5686 Rel wrel 5689 Fun wfun 6554 defAt wdfat 47133 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-fun 6562 df-dfat 47136 | 
| This theorem is referenced by: dfafv2 47149 afveu 47170 rlimdmafv 47194 tz6.12-2-afv2 47254 afv2eu 47255 tz6.12i-afv2 47260 dfatbrafv2b 47262 dfatsnafv2 47269 dfafv23 47270 dfatcolem 47272 dfatco 47273 rlimdmafv2 47275 | 
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