Theorem dffunsALTV4 39703

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 39700	. 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2		cosselcnvrefrels4 39552	. . 3 ⊢ ( ≀ 𝑓 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑓𝑥 ∧ ≀ 𝑓 ∈ Rels ))
3		cosselrels 39507	. . . 4 ⊢ (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels )
4	3	biantrud 541	. . 3 ⊢ (𝑓 ∈ Rels → (∀𝑢∃*𝑥 𝑢𝑓𝑥 ↔ (∀𝑢∃*𝑥 𝑢𝑓𝑥 ∧ ≀ 𝑓 ∈ Rels )))
5	2, 4	bitr4id 293	. 2 ⊢ (𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥))
6	1, 5	rabimbieq 39185	1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}

This proof depends on syntax axioms:   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃*wmo 2563  {crab 3413   class class class wbr 5103   ≀ ccoss 39115   Rels crels 39117   CnvRefRels ccnvrefrels 39123   FunsALTV cfunsALTV 39147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-rels 39372  df-coss 39433  df-ssr 39510  df-cnvrefs 39537  df-cnvrefrels 39538  df-funss 39697  df-funsALTV 39698

This theorem is used by:  dffunsALTV5  39704