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Theorem tposeq 8163
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 4004 . . 3 (𝐹 = 𝐺𝐹𝐺)
2 tposss 8162 . . 3 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
31, 2syl 17 . 2 (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
4 eqimss2 4005 . . 3 (𝐹 = 𝐺𝐺𝐹)
5 tposss 8162 . . 3 (𝐺𝐹 → tpos 𝐺 ⊆ tpos 𝐹)
64, 5syl 17 . 2 (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹)
73, 6eqssd 3965 1 (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wss 3914  tpos ctpos 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-mpt 5193  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-res 5649  df-tpos 8161
This theorem is referenced by:  tposeqd  8164  tposeqi  8194
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