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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 6148 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))) | |
2 | nfcv 2934 | . . . . . . 7 ⊢ Ⅎ𝑦𝑤 | |
3 | dffun6f.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2934 | . . . . . . 7 ⊢ Ⅎ𝑦𝑣 | |
5 | 2, 3, 4 | nfbr 4935 | . . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣 |
6 | nfv 1957 | . . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦 | |
7 | breq2 4892 | . . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦)) | |
8 | 5, 6, 7 | cbvmo 2637 | . . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦) |
9 | 8 | albii 1863 | . . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦) |
10 | df-mo 2551 | . . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)) | |
11 | 10 | albii 1863 | . . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)) |
12 | nfcv 2934 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
13 | dffun6f.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
14 | nfcv 2934 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
15 | 12, 13, 14 | nfbr 4935 | . . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦 |
16 | 15 | nfmo 2577 | . . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦 |
17 | nfv 1957 | . . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦 | |
18 | breq1 4891 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
19 | 18 | mobidv 2564 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦)) |
20 | 16, 17, 19 | cbvalv1 2312 | . . . 4 ⊢ (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
21 | 9, 11, 20 | 3bitr3ri 294 | . . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)) |
22 | 21 | anbi2i 616 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))) |
23 | 1, 22 | bitr4i 270 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∀wal 1599 ∃wex 1823 ∃*wmo 2549 Ⅎwnfc 2919 class class class wbr 4888 Rel wrel 5362 Fun wfun 6131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-id 5263 df-cnv 5365 df-co 5366 df-fun 6139 |
This theorem is referenced by: dffun6 6152 funopab 6172 funcnvmpt 30036 dffun3f 43544 |
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