Theorem dffunsALTV5 39747

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

Assertion

Ref	Expression
dffunsALTV5	⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)}

Distinct variable group:   𝑥,𝑓,𝑦

Proof of Theorem dffunsALTV5

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffunsALTV4 39746	. 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
2		ineccnvmo2 39343	. . 3 ⊢ (∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥)
3	2	rabbii 3417	. 2 ⊢ {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)} = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
4	1, 3	eqtr4i 2786	1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)}

This proof depends on syntax axioms:   ∨ wo 861  ∀wal 1567   = wceq 1569  ∃*wmo 2562  ∀wral 3076  {crab 3412   ∩ cin 4089  ∅c0 4278   class class class wbr 5102  ^◡ccnv 5649  ran crn 5651  [cec 8715   Rels crels 39160   FunsALTV cfunsALTV 39190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8719  df-rels 39415  df-coss 39476  df-ssr 39553  df-cnvrefs 39580  df-cnvrefrels 39581  df-funss 39740  df-funsALTV 39741

This theorem is used by: (None)