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Mirrors > Home > MPE Home > Th. List > dffun6 | Structured version Visualization version GIF version |
Description: Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2138, ax-12 2172. (Revised by SN, 19-Dec-2024.) |
Ref | Expression |
---|---|
dffun6 | ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun2 6554 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))) | |
2 | breq2 5153 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑥𝐹𝑧)) | |
3 | 2 | mo4 2561 | . . . 4 ⊢ (∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)) |
4 | 3 | albii 1822 | . . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)) |
5 | 4 | anbi2i 624 | . 2 ⊢ ((Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦) ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))) |
6 | 1, 5 | bitr4i 278 | 1 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 ∃*wmo 2533 class class class wbr 5149 Rel wrel 5682 Fun wfun 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-fun 6546 |
This theorem is referenced by: dffun3 6558 funmo 6564 funmoOLD 6565 dffun7 6576 fununfun 6597 funcnvsn 6599 funcnv2 6617 svrelfun 6621 funimaexg 6635 fnres 6678 nfunsn 6934 dff3 7102 brdom3 10523 nqerf 10925 shftfn 15020 cnextfun 23568 perfdvf 25420 taylf 25873 funressnvmo 45755 |
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