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Mirrors > Home > MPE Home > Th. List > Mathboxes > elfunsALTVfunALTV | Structured version Visualization version GIF version |
Description: The element of the class of functions and the function predicate are the same when 𝐹 is a set. (Contributed by Peter Mazsa, 26-Jul-2021.) |
Ref | Expression |
---|---|
elfunsALTVfunALTV | ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossex 35697 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → ≀ 𝐹 ∈ V) | |
2 | elcnvrefrelsrel 35805 | . . . 4 ⊢ ( ≀ 𝐹 ∈ V → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝑉 → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹)) |
4 | elrelsrel 35760 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ Rels ↔ Rel 𝐹)) | |
5 | 3, 4 | anbi12d 632 | . 2 ⊢ (𝐹 ∈ 𝑉 → (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))) |
6 | elfunsALTV 35958 | . 2 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) | |
7 | df-funALTV 35948 | . 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | |
8 | 5, 6, 7 | 3bitr4g 316 | 1 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 Vcvv 3491 Rel wrel 5553 ≀ ccoss 35486 Rels crels 35488 CnvRefRels ccnvrefrels 35494 CnvRefRel wcnvrefrel 35495 FunsALTV cfunsALTV 35516 FunALTV wfunALTV 35517 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-coss 35692 df-rels 35758 df-ssr 35771 df-cnvrefs 35796 df-cnvrefrels 35797 df-cnvrefrel 35798 df-funss 35946 df-funsALTV 35947 df-funALTV 35948 |
This theorem is referenced by: (None) |
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