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| Mirrors > Home > MPE Home > Th. List > dffun8 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6545. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
| Ref | Expression |
|---|---|
| dffun8 | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dffun7 6545 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)) | |
| 2 | moeu 2577 | . . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦)) | |
| 3 | vex 3454 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 4 | 3 | eldm 5866 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦) |
| 5 | pm5.5 361 | . . . . . 6 ⊢ (∃𝑦 𝑥𝐴𝑦 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦)) | |
| 6 | 4, 5 | sylbi 217 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐴 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦)) |
| 7 | 2, 6 | bitrid 283 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 → (∃*𝑦 𝑥𝐴𝑦 ↔ ∃!𝑦 𝑥𝐴𝑦)) |
| 8 | 7 | ralbiia 3074 | . . 3 ⊢ (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦) |
| 9 | 8 | anbi2i 623 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦)) |
| 10 | 1, 9 | bitri 275 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2109 ∃*wmo 2532 ∃!weu 2562 ∀wral 3045 class class class wbr 5109 dom cdm 5640 Rel wrel 5645 Fun wfun 6507 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pr 5389 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-ss 3933 df-nul 4299 df-if 4491 df-sn 4592 df-pr 4594 df-op 4598 df-br 5110 df-opab 5172 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-fun 6515 |
| This theorem is referenced by: dfdfat2 47119 |
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