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| Mirrors > Home > MPE Home > Th. List > dffun9 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) | 
| Ref | Expression | 
|---|---|
| dffun9 | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dffun7 6592 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)) | |
| 2 | vex 3483 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 3 | vex 3483 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
| 4 | 2, 3 | brelrn 5952 | . . . . . . 7 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴) | 
| 5 | 4 | pm4.71ri 560 | . . . . . 6 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) | 
| 6 | 5 | mobii 2547 | . . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) | 
| 7 | df-rmo 3379 | . . . . 5 ⊢ (∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) | |
| 8 | 6, 7 | bitr4i 278 | . . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦) | 
| 9 | 8 | ralbii 3092 | . . 3 ⊢ (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦) | 
| 10 | 9 | anbi2i 623 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)) | 
| 11 | 1, 10 | bitri 275 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ∃*wmo 2537 ∀wral 3060 ∃*wrmo 3378 class class class wbr 5142 dom cdm 5684 ran crn 5685 Rel wrel 5689 Fun wfun 6554 | 
| 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-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rmo 3379 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 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-rn 5695 df-fun 6562 | 
| This theorem is referenced by: brdom4 10571 | 
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