Theorem tposeq 6478

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 3292	. . 3 |- ( F = G -> F C_ G )
2		tposss 6477	. . 3 |- ( F C_ G -> tpos F C_ tpos G )
3	1, 2	syl 14	. 2 |- ( F = G -> tpos F C_ tpos G )
4		eqimss2 3293	. . 3 |- ( F = G -> G C_ F )
5		tposss 6477	. . 3 |- ( G C_ F -> tpos G C_ tpos F )
6	4, 5	syl 14	. 2 |- ( F = G -> tpos G C_ tpos F )
7	3, 6	eqssd 3255	1 |- ( F = G -> tpos F = tpos G )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3211  tpos ctpos 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-mpt 4173  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-tpos 6476

This theorem is referenced by:  tposeqd  6479  tposeqi  6508