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Theorem tposeq 6265
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 3223 . . 3 (𝐹 = 𝐺𝐹𝐺)
2 tposss 6264 . . 3 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
31, 2syl 14 . 2 (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
4 eqimss2 3224 . . 3 (𝐹 = 𝐺𝐺𝐹)
5 tposss 6264 . . 3 (𝐺𝐹 → tpos 𝐺 ⊆ tpos 𝐹)
64, 5syl 14 . 2 (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹)
73, 6eqssd 3186 1 (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wss 3143  tpos ctpos 6262
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-mpt 4080  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-res 4652  df-tpos 6263
This theorem is referenced by:  tposeqd  6266  tposeqi  6295
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