Theorem elfunsALTV3 39226

Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)

Assertion

Ref	Expression
elfunsALTV3	⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ 𝐹 ∈ Rels ))

Distinct variable group:   𝑢,𝐹,𝑥,𝑦

Proof of Theorem elfunsALTV3

Step	Hyp	Ref	Expression
1		elfunsALTV 39224	. 2 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
2		cosselcnvrefrels3 39066	. . . 4 ⊢ ( ≀ 𝐹 ∈ CnvRefRels ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝐹 ∈ Rels ))
3		cosselrels 39022	. . . . 5 ⊢ (𝐹 ∈ Rels → ≀ 𝐹 ∈ Rels )
4	3	biantrud 538	. . . 4 ⊢ (𝐹 ∈ Rels → (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝐹 ∈ Rels )))
5	2, 4	bitr4id 292	. . 3 ⊢ (𝐹 ∈ Rels → ( ≀ 𝐹 ∈ CnvRefRels ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦)))
6	5	pm5.32ri 582	. 2 ⊢ (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ 𝐹 ∈ Rels ))
7	1, 6	bitri 277	1 ⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ 𝐹 ∈ Rels ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1552   = wceq 1554   ∈ wcel 2136   class class class wbr 5094   ≀ ccoss 38630   Rels crels 38632   CnvRefRels ccnvrefrels 38638   FunsALTV cfunsALTV 38662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-rels 38887  df-coss 38948  df-ssr 39025  df-cnvrefs 39052  df-cnvrefrels 39053  df-funss 39212  df-funsALTV 39213

This theorem is referenced by: (None)