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Mirrors > Home > MPE Home > Th. List > Mathboxes > dffunsALTV2 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.) |
Ref | Expression |
---|---|
dffunsALTV2 | ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffunsALTV 38588 | . 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels } | |
2 | cosselcnvrefrels2 38443 | . . 3 ⊢ ( ≀ 𝑓 ∈ CnvRefRels ↔ ( ≀ 𝑓 ⊆ I ∧ ≀ 𝑓 ∈ Rels )) | |
3 | cosselrels 38401 | . . . 4 ⊢ (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels ) | |
4 | 3 | biantrud 531 | . . 3 ⊢ (𝑓 ∈ Rels → ( ≀ 𝑓 ⊆ I ↔ ( ≀ 𝑓 ⊆ I ∧ ≀ 𝑓 ∈ Rels ))) |
5 | 2, 4 | bitr4id 290 | . 2 ⊢ (𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ≀ 𝑓 ⊆ I )) |
6 | 1, 5 | rabimbieq 38156 | 1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I } |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2103 {crab 3438 ⊆ wss 3970 I cid 5596 ≀ ccoss 38084 Rels crels 38086 CnvRefRels ccnvrefrels 38092 FunsALTV cfunsALTV 38114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-coss 38316 df-rels 38390 df-ssr 38403 df-cnvrefs 38430 df-cnvrefrels 38431 df-funss 38585 df-funsALTV 38586 |
This theorem is referenced by: (None) |
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