Theorem dffunsALTV5 35954

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 35953	. 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
2		ineccnvmo2 35648	. . 3 ⊢ (∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥)
3	2	rabbii 3470	. 2 ⊢ {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)} = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
4	1, 3	eqtr4i 2846	1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)}

Syntax hints:   ∨ wo 843  ∀wal 1534   = wceq 1536  ∃*wmo 2619  ∀wral 3137  {crab 3141   ∩ cin 3928  ∅c0 4284   class class class wbr 5059  ^◡ccnv 5547  ran crn 5549  [cec 8280   Rels crels 35489   FunsALTV cfunsALTV 35517

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-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  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-ima 5561  df-ec 8284  df-coss 35693  df-rels 35759  df-ssr 35772  df-cnvrefs 35797  df-cnvrefrels 35798  df-funss 35947  df-funsALTV 35948

This theorem is referenced by: (None)