Theorem dffunsALTV3 39439

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 ∀ y1 ∀ y2 (( x1 𝑓 y1 ∧ x1 𝑓 y2 ) → y1 = y2 )}. (Contributed by Peter Mazsa, 30-Aug-2021.)

Assertion

Ref	Expression
dffunsALTV3	⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦)}

Distinct variable group:   𝑢,𝑓,𝑥,𝑦

Proof of Theorem dffunsALTV3

Step	Hyp	Ref	Expression
1		dffunsALTV 39437	. 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2		cosselcnvrefrels3 39288	. . 3 ⊢ ( ≀ 𝑓 ∈ CnvRefRels ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑓 ∈ Rels ))
3		cosselrels 39244	. . . 4 ⊢ (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels )
4	3	biantrud 540	. . 3 ⊢ (𝑓 ∈ Rels → (∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦) ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑓 ∈ Rels )))
5	2, 4	bitr4id 293	. 2 ⊢ (𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦)))
6	1, 5	rabimbieq 38922	1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦)}

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570   ∈ wcel 2143  {crab 3416   class class class wbr 5109   ≀ ccoss 38852   Rels crels 38854   CnvRefRels ccnvrefrels 38860   FunsALTV cfunsALTV 38884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-rels 39109  df-coss 39170  df-ssr 39247  df-cnvrefs 39274  df-cnvrefrels 39275  df-funss 39434  df-funsALTV 39435

This theorem is referenced by: (None)