Theorem cureq 33385

Description: Equality theorem for currying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
cureq	|- ( A = B -> curry A = curry B )

Proof of Theorem cureq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5324	. . . 4 |- ( A = B -> dom A = dom B )
2	1	dmeqd 5326	. . 3 |- ( A = B -> dom dom A = dom dom B )
3		breq 4655	. . . 4 |- ( A = B -> ( <. x , y >. A z <-> <. x , y >. B z ) )
4	3	opabbidv 4716	. . 3 |- ( A = B -> { <. y , z >. | <. x , y >. A z } = { <. y , z >. | <. x , y >. B z } )
5	2, 4	mpteq12dv 4733	. 2 |- ( A = B -> ( x e. dom dom A |-> { <. y , z >. | <. x , y >. A z } ) = ( x e. dom dom B |-> { <. y , z >. | <. x , y >. B z } ) )
6		df-cur 7393	. 2 |- curry A = ( x e. dom dom A |-> { <. y , z >. | <. x , y >. A z } )
7		df-cur 7393	. 2 |- curry B = ( x e. dom dom B |-> { <. y , z >. | <. x , y >. B z } )
8	5, 6, 7	3eqtr4g 2681	1 |- ( A = B -> curry A = curry B )

Syntax hints:   -> wi 4   = wceq 1483   <.cop 4183   class class class wbr 4653   {copab 4712   |-> cmpt 4729   dom cdm 5114  curry ccur 7391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-dm 5124  df-cur 7393

This theorem is referenced by:  curfv  33389  matunitlindf  33407