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Mirrors > Home > MPE Home > Th. List > Mathboxes > elfunsALTV4 | Structured version Visualization version GIF version |
Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.) |
Ref | Expression |
---|---|
elfunsALTV4 | ⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ 𝐹 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfunsALTV 38673 | . 2 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) | |
2 | cosselcnvrefrels4 38521 | . . . 4 ⊢ ( ≀ 𝐹 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ ≀ 𝐹 ∈ Rels )) | |
3 | cosselrels 38477 | . . . . 5 ⊢ (𝐹 ∈ Rels → ≀ 𝐹 ∈ Rels ) | |
4 | 3 | biantrud 531 | . . . 4 ⊢ (𝐹 ∈ Rels → (∀𝑢∃*𝑥 𝑢𝐹𝑥 ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ ≀ 𝐹 ∈ Rels ))) |
5 | 2, 4 | bitr4id 290 | . . 3 ⊢ (𝐹 ∈ Rels → ( ≀ 𝐹 ∈ CnvRefRels ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)) |
6 | 5 | pm5.32ri 575 | . 2 ⊢ (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ 𝐹 ∈ Rels )) |
7 | 1, 6 | bitri 275 | 1 ⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ 𝐹 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∀wal 1534 ∈ wcel 2105 ∃*wmo 2535 class class class wbr 5147 ≀ ccoss 38161 Rels crels 38163 CnvRefRels ccnvrefrels 38169 FunsALTV cfunsALTV 38191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-coss 38392 df-rels 38466 df-ssr 38479 df-cnvrefs 38506 df-cnvrefrels 38507 df-funss 38661 df-funsALTV 38662 |
This theorem is referenced by: (None) |
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