Theorem dffunsALTV4 35952

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.)

Assertion

Ref	Expression
dffunsALTV4	⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}

Distinct variable group:   𝑢,𝑓,𝑥

Proof of Theorem dffunsALTV4

Step	Hyp	Ref	Expression
1		dffunsALTV 35949	. 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2		cosselrels 35769	. . . 4 ⊢ (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels )
3	2	biantrud 534	. . 3 ⊢ (𝑓 ∈ Rels → (∀𝑢∃*𝑥 𝑢𝑓𝑥 ↔ (∀𝑢∃*𝑥 𝑢𝑓𝑥 ∧ ≀ 𝑓 ∈ Rels )))
4		cosselcnvrefrels4 35809	. . 3 ⊢ ( ≀ 𝑓 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑓𝑥 ∧ ≀ 𝑓 ∈ Rels ))
5	3, 4	syl6rbbr 292	. 2 ⊢ (𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥))
6	1, 5	rabimbieq 35546	1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}

Syntax hints:   ∧ wa 398  ∀wal 1534   = wceq 1536   ∈ wcel 2113  ∃*wmo 2619  {crab 3141   class class class wbr 5059   ≀ ccoss 35486   Rels crels 35488   CnvRefRels ccnvrefrels 35494   FunsALTV cfunsALTV 35516

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-id 5453  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-funss 35946  df-funsALTV 35947

This theorem is referenced by:  dffunsALTV5  35953