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Mathbox for Emmett Weisz |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dffun3f | Structured version Visualization version GIF version |
Description: Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.) |
Ref | Expression |
---|---|
dffun3f.1 | ⊢ Ⅎ𝑥𝐴 |
dffun3f.2 | ⊢ Ⅎ𝑦𝐴 |
dffun3f.3 | ⊢ Ⅎ𝑧𝐴 |
Ref | Expression |
---|---|
dffun3f | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun3f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | dffun3f.2 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
3 | 1, 2 | dffun6f 6137 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) |
4 | nfcv 2969 | . . . . . 6 ⊢ Ⅎ𝑧𝑥 | |
5 | dffun3f.3 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
6 | nfcv 2969 | . . . . . 6 ⊢ Ⅎ𝑧𝑦 | |
7 | 4, 5, 6 | nfbr 4920 | . . . . 5 ⊢ Ⅎ𝑧 𝑥𝐴𝑦 |
8 | 7 | mof 2631 | . . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) |
9 | 8 | albii 1920 | . . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) |
10 | 9 | anbi2i 618 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
11 | 3, 10 | bitri 267 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∀wal 1656 ∃wex 1880 ∃*wmo 2603 Ⅎwnfc 2956 class class class wbr 4873 Rel wrel 5347 Fun wfun 6117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-id 5250 df-cnv 5350 df-co 5351 df-fun 6125 |
This theorem is referenced by: setrec2lem2 43336 |
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