Theorem tposeq 7354

Description: Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
tposeq	|- ( F = G -> tpos F = tpos G )

Proof of Theorem tposeq

Step	Hyp	Ref	Expression
1		eqimss 3657	. . 3 |- ( F = G -> F C_ G )
2		tposss 7353	. . 3 |- ( F C_ G -> tpos F C_ tpos G )
3	1, 2	syl 17	. 2 |- ( F = G -> tpos F C_ tpos G )
4		eqimss2 3658	. . 3 |- ( F = G -> G C_ F )
5		tposss 7353	. . 3 |- ( G C_ F -> tpos G C_ tpos F )
6	4, 5	syl 17	. 2 |- ( F = G -> tpos G C_ tpos F )
7	3, 6	eqssd 3620	1 |- ( F = G -> tpos F = tpos G )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574  tpos ctpos 7351

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-tpos 7352

This theorem is referenced by:  tposeqd  7355  tposeqi  7385