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| Mirrors > Home > MPE Home > Th. List > dffun6f | Structured version Visualization version GIF version | ||
| Description: Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
| Ref | Expression |
|---|---|
| dffun6f.1 | ⊢ Ⅎ𝑥𝐴 |
| dffun6f.2 | ⊢ Ⅎ𝑦𝐴 |
| Ref | Expression |
|---|---|
| dffun6f | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dffun3 6575 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))) | |
| 2 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑦𝑤 | |
| 3 | dffun6f.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐴 | |
| 4 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑦𝑣 | |
| 5 | 2, 3, 4 | nfbr 5190 | . . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣 |
| 6 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦 | |
| 7 | breq2 5147 | . . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦)) | |
| 8 | 5, 6, 7 | cbvmow 2603 | . . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦) |
| 9 | 8 | albii 1819 | . . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦) |
| 10 | df-mo 2540 | . . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)) | |
| 11 | 10 | albii 1819 | . . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)) |
| 12 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
| 13 | dffun6f.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
| 14 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
| 15 | 12, 13, 14 | nfbr 5190 | . . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦 |
| 16 | 15 | nfmov 2560 | . . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦 |
| 17 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦 | |
| 18 | breq1 5146 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
| 19 | 18 | mobidv 2549 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦)) |
| 20 | 16, 17, 19 | cbvalv1 2343 | . . . 4 ⊢ (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
| 21 | 9, 11, 20 | 3bitr3ri 302 | . . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)) |
| 22 | 21 | anbi2i 623 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))) |
| 23 | 1, 22 | bitr4i 278 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∃wex 1779 ∃*wmo 2538 Ⅎwnfc 2890 class class class wbr 5143 Rel wrel 5690 Fun wfun 6555 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 |
| This theorem is referenced by: dffun6OLD 6580 funopab 6601 funcnvmpt 32677 dffun3f 49201 |
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