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Mirrors > Home > MPE Home > Th. List > Mathboxes > dffunsALTV3 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of functions. For the 𝑋 axis and the 𝑌 axis you can convert the right side to {𝑓 ∈ Rels ∣ ∀ x1 ∀ y1 ∀ y2 (( x1 𝑓 y1 ∧ x1 𝑓 y2 ) → y1 = y2 )}. (Contributed by Peter Mazsa, 30-Aug-2021.) |
Ref | Expression |
---|---|
dffunsALTV3 | ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffunsALTV 38665 | . 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels } | |
2 | cosselcnvrefrels3 38521 | . . 3 ⊢ ( ≀ 𝑓 ∈ CnvRefRels ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑓 ∈ Rels )) | |
3 | cosselrels 38478 | . . . 4 ⊢ (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels ) | |
4 | 3 | biantrud 531 | . . 3 ⊢ (𝑓 ∈ Rels → (∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦) ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑓 ∈ Rels ))) |
5 | 2, 4 | bitr4id 290 | . 2 ⊢ (𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦))) |
6 | 1, 5 | rabimbieq 38233 | 1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 {crab 3433 class class class wbr 5148 ≀ ccoss 38162 Rels crels 38164 CnvRefRels ccnvrefrels 38170 FunsALTV cfunsALTV 38192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-coss 38393 df-rels 38467 df-ssr 38480 df-cnvrefs 38507 df-cnvrefrels 38508 df-funss 38662 df-funsALTV 38663 |
This theorem is referenced by: (None) |
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