Theorem dffunsALTV5 38733

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 38732	. 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
2		ineccnvmo2 38396	. . 3 ⊢ (∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥)
3	2	rabbii 3400	. 2 ⊢ {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)} = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
4	1, 3	eqtr4i 2757	1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)}

Syntax hints:   ∨ wo 847  ∀wal 1539   = wceq 1541  ∃*wmo 2533  ∀wral 3047  {crab 3395   ∩ cin 3896  ∅c0 4280   class class class wbr 5089  ^◡ccnv 5613  ran crn 5615  [cec 8620   Rels crels 38225   FunsALTV cfunsALTV 38253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624  df-coss 38456  df-rels 38530  df-ssr 38543  df-cnvrefs 38570  df-cnvrefrels 38571  df-funss 38726  df-funsALTV 38727

This theorem is referenced by: (None)