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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 6557 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))) | |
2 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑦𝑤 | |
3 | dffun6f.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑦𝑣 | |
5 | 2, 3, 4 | nfbr 5195 | . . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣 |
6 | nfv 1916 | . . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦 | |
7 | breq2 5152 | . . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦)) | |
8 | 5, 6, 7 | cbvmow 2596 | . . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦) |
9 | 8 | albii 1820 | . . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦) |
10 | df-mo 2533 | . . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)) | |
11 | 10 | albii 1820 | . . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)) |
12 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
13 | dffun6f.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
14 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
15 | 12, 13, 14 | nfbr 5195 | . . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦 |
16 | 15 | nfmov 2553 | . . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦 |
17 | nfv 1916 | . . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦 | |
18 | breq1 5151 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
19 | 18 | mobidv 2542 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦)) |
20 | 16, 17, 19 | cbvalv1 2336 | . . . 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 622 | . 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 205 ∧ wa 395 ∀wal 1538 ∃wex 1780 ∃*wmo 2531 Ⅎwnfc 2882 class class class wbr 5148 Rel wrel 5681 Fun wfun 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-fun 6545 |
This theorem is referenced by: dffun6OLD 6562 funopab 6583 funcnvmpt 32326 dffun3f 47891 |
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