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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 6045 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) |
4 | nfcv 2913 | . . . . . 6 ⊢ Ⅎ𝑧𝑥 | |
5 | dffun3f.3 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
6 | nfcv 2913 | . . . . . 6 ⊢ Ⅎ𝑧𝑦 | |
7 | 4, 5, 6 | nfbr 4833 | . . . . 5 ⊢ Ⅎ𝑧 𝑥𝐴𝑦 |
8 | 7 | mo2 2627 | . . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) |
9 | 8 | albii 1895 | . . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) |
10 | 9 | anbi2i 609 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
11 | 3, 10 | bitri 264 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∀wal 1629 ∃wex 1852 ∃*wmo 2619 Ⅎwnfc 2900 class class class wbr 4786 Rel wrel 5254 Fun wfun 6025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-id 5157 df-cnv 5257 df-co 5258 df-fun 6033 |
This theorem is referenced by: setrec2lem2 42969 |
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