Theorem dffunsALTV5 39970

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 39969	. 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
2		ineccnvmo2 39566	. . 3 ⊢ (∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥)
3	2	rabbii 3421	. 2 ⊢ {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)} = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
4	1, 3	eqtr4i 2789	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 1568   = wceq 1570  ∃*wmo 2565  ∀wral 3080  {crab 3416   ∩ cin 4092  ∅c0 4281   class class class wbr 5108  ^◡ccnv 5657  ran crn 5659  [cec 8753   Rels crels 39383   FunsALTV cfunsALTV 39413

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-rels 39638  df-coss 39699  df-ssr 39776  df-cnvrefs 39803  df-cnvrefrels 39804  df-funss 39963  df-funsALTV 39964

This theorem is used by: (None)