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Mirrors > Home > ILE Home > Th. List > dffun6f | 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 | dffun2 4912 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))) | |
2 | nfcv 2178 | . . . . . . 7 ⊢ Ⅎ𝑦𝑤 | |
3 | dffun6f.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2178 | . . . . . . 7 ⊢ Ⅎ𝑦𝑣 | |
5 | 2, 3, 4 | nfbr 3808 | . . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣 |
6 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦 | |
7 | breq2 3768 | . . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦)) | |
8 | 5, 6, 7 | cbvmo 1940 | . . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦) |
9 | 8 | albii 1359 | . . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦) |
10 | breq2 3768 | . . . . . 6 ⊢ (𝑣 = 𝑢 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑢)) | |
11 | 10 | mo4 1961 | . . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢)) |
12 | 11 | albii 1359 | . . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢)) |
13 | nfcv 2178 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
14 | dffun6f.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
15 | nfcv 2178 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
16 | 13, 14, 15 | nfbr 3808 | . . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦 |
17 | 16 | nfmo 1920 | . . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦 |
18 | nfv 1421 | . . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦 | |
19 | breq1 3767 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
20 | 19 | mobidv 1936 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦)) |
21 | 17, 18, 20 | cbval 1637 | . . . 4 ⊢ (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
22 | 9, 12, 21 | 3bitr3ri 200 | . . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢)) |
23 | 22 | anbi2i 430 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))) |
24 | 1, 23 | bitr4i 176 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 ∃*wmo 1901 Ⅎwnfc 2165 class class class wbr 3764 Rel wrel 4350 Fun wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-cnv 4353 df-co 4354 df-fun 4904 |
This theorem is referenced by: dffun6 4916 dffun4f 4918 funopab 4935 |
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