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

Step	Hyp	Ref	Expression
1		eqimss 4042	. . 3 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺)
2		tposss 8252	. . 3 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
3	1, 2	syl 17	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
4		eqimss2 4043	. . 3 ⊢ (𝐹 = 𝐺 → 𝐺 ⊆ 𝐹)
5		tposss 8252	. . 3 ⊢ (𝐺 ⊆ 𝐹 → tpos 𝐺 ⊆ tpos 𝐹)
6	4, 5	syl 17	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹)
7	3, 6	eqssd 4001	1 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)

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