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Theorem tposeq 8253
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
StepHypRef Expression
1 eqimss 4042 . . 3 (𝐹 = 𝐺𝐹𝐺)
2 tposss 8252 . . 3 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
31, 2syl 17 . 2 (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
4 eqimss2 4043 . . 3 (𝐹 = 𝐺𝐺𝐹)
5 tposss 8252 . . 3 (𝐺𝐹 → tpos 𝐺 ⊆ tpos 𝐹)
64, 5syl 17 . 2 (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹)
73, 6eqssd 4001 1 (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wss 3951  tpos ctpos 8250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-tpos 8251
This theorem is referenced by:  tposeqd  8254  tposeqi  8284
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