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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dffunsALTV4 | 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∃*𝑦1𝑥1𝑓𝑦1}. (Contributed by Peter Mazsa, 31-Aug-2021.) | 
| Ref | Expression | 
|---|---|
| dffunsALTV4 | ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dffunsALTV 38684 | . 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels } | |
| 2 | cosselcnvrefrels4 38541 | . . 3 ⊢ ( ≀ 𝑓 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑓𝑥 ∧ ≀ 𝑓 ∈ Rels )) | |
| 3 | cosselrels 38497 | . . . 4 ⊢ (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels ) | |
| 4 | 3 | biantrud 531 | . . 3 ⊢ (𝑓 ∈ Rels → (∀𝑢∃*𝑥 𝑢𝑓𝑥 ↔ (∀𝑢∃*𝑥 𝑢𝑓𝑥 ∧ ≀ 𝑓 ∈ Rels ))) | 
| 5 | 2, 4 | bitr4id 290 | . 2 ⊢ (𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥)) | 
| 6 | 1, 5 | rabimbieq 38252 | 1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃*wmo 2538 {crab 3436 class class class wbr 5143 ≀ ccoss 38182 Rels crels 38184 CnvRefRels ccnvrefrels 38190 FunsALTV cfunsALTV 38212 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-coss 38412 df-rels 38486 df-ssr 38499 df-cnvrefs 38526 df-cnvrefrels 38527 df-funss 38681 df-funsALTV 38682 | 
| This theorem is referenced by: dffunsALTV5 38688 | 
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