Theorem bj-projeq 32980

Description: Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-projeq	|- ( A = C -> ( B = D -> ( A Proj B ) = ( C Proj D ) ) )

Proof of Theorem bj-projeq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( A = C /\ B = D ) -> B = D )
2		simpl 473	. . . . . . 7 |- ( ( A = C /\ B = D ) -> A = C )
3	2	sneqd 4189	. . . . . 6 |- ( ( A = C /\ B = D ) -> { A } = { C } )
4	1, 3	imaeq12d 5467	. . . . 5 |- ( ( A = C /\ B = D ) -> ( B " { A } ) = ( D " { C } ) )
5	4	eleq2d 2687	. . . 4 |- ( ( A = C /\ B = D ) -> ( { x } e. ( B " { A } ) <-> { x } e. ( D " { C } ) ) )
6	5	abbidv 2741	. . 3 |- ( ( A = C /\ B = D ) -> { x | { x } e. ( B " { A } ) } = { x | { x } e. ( D " { C } ) } )
7		df-bj-proj 32979	. . 3 |- ( A Proj B ) = { x | { x } e. ( B " { A } ) }
8		df-bj-proj 32979	. . 3 |- ( C Proj D ) = { x | { x } e. ( D " { C } ) }
9	6, 7, 8	3eqtr4g 2681	. 2 |- ( ( A = C /\ B = D ) -> ( A Proj B ) = ( C Proj D ) )
10	9	ex 450	1 |- ( A = C -> ( B = D -> ( A Proj B ) = ( C Proj D ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {csn 4177   "cima 5117   Proj bj-cproj 32978

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-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-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-bj-proj 32979

This theorem is referenced by:  bj-projeq2  32981