Theorem fungid 8272

Description: Id is a function. (Contributed by FL, 5-Feb-2010.)

Assertion

Ref	Expression
fungid	⊢ Fun Id

Proof of Theorem fungid

Step	Hyp	Ref	Expression
1		funopabeq 3654	. 2 ⊢ Fun {<.w, y>.∣y = ∪{z ∈ ran w∣∀x ∈ ran w((zwx) = x ⋀ (xwz) = x)}}
2		df-gid 8250	. . 3 ⊢ Id = {<.w, y>.∣y = ∪{z ∈ ran w∣∀x ∈ ran w((zwx) = x ⋀ (xwz) = x)}}
3		funeq 3640	. . 3 ⊢ (Id = {<.w, y>.∣y = ∪{z ∈ ran w∣∀x ∈ ran w((zwx) = x ⋀ (xwz) = x)}} → (Fun Id ↔ Fun {<.w, y>.∣y = ∪{z ∈ ran w∣∀x ∈ ran w((zwx) = x ⋀ (xwz) = x)}}))
4	2, 3	ax-mp 7	. 2 ⊢ (Fun Id ↔ Fun {<.w, y>.∣y = ∪{z ∈ ran w∣∀x ∈ ran w((zwx) = x ⋀ (xwz) = x)}})
5	1, 4	mpbir 188	1 ⊢ Fun Id

Syntax hints:   ↔ wb 144   ⋀ wa 221   = wceq 992  ∀wral 1691  {crab 1694  ∪cuni 2569  {copab 2740  ran crn 3252  Fun wfun 3257  (class class class)co 4021  Idcgi 8245

This theorem is referenced by:  0vfval 8472

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-fun 3273  df-gid 8250

Copyright terms: Public domain