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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 6554 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) |
4 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑧𝑥 | |
5 | dffun3f.3 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
6 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑧𝑦 | |
7 | 4, 5, 6 | nfbr 5188 | . . . . 5 ⊢ Ⅎ𝑧 𝑥𝐴𝑦 |
8 | 7 | mof 2551 | . . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) |
9 | 8 | albii 1813 | . . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) |
10 | 9 | anbi2i 622 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
11 | 3, 10 | bitri 275 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 ∃wex 1773 ∃*wmo 2526 Ⅎwnfc 2877 class class class wbr 5141 Rel wrel 5674 Fun wfun 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-fun 6538 |
This theorem is referenced by: setrec2lem2 47994 |
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