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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 6394 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) |
4 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑧𝑥 | |
5 | dffun3f.3 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
6 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑧𝑦 | |
7 | 4, 5, 6 | nfbr 5100 | . . . . 5 ⊢ Ⅎ𝑧 𝑥𝐴𝑦 |
8 | 7 | mof 2562 | . . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) |
9 | 8 | albii 1827 | . . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) |
10 | 9 | anbi2i 626 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
11 | 3, 10 | bitri 278 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 ∃wex 1787 ∃*wmo 2537 Ⅎwnfc 2884 class class class wbr 5053 Rel wrel 5556 Fun wfun 6374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-id 5455 df-cnv 5559 df-co 5560 df-fun 6382 |
This theorem is referenced by: setrec2lem2 46071 |
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