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Theorem tposeq 7505
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 3804 . . 3 (𝐹 = 𝐺𝐹𝐺)
2 tposss 7504 . . 3 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
31, 2syl 17 . 2 (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
4 eqimss2 3805 . . 3 (𝐹 = 𝐺𝐺𝐹)
5 tposss 7504 . . 3 (𝐺𝐹 → tpos 𝐺 ⊆ tpos 𝐹)
64, 5syl 17 . 2 (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹)
73, 6eqssd 3767 1 (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wss 3721  tpos ctpos 7502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-mpt 4862  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-tpos 7503
This theorem is referenced by:  tposeqd  7506  tposeqi  7536
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