Theorem tposeq 8171

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 3981	. . 3 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺)
2		tposss 8170	. . 3 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
3	1, 2	syl 17	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
4		eqimss2 3982	. . 3 ⊢ (𝐹 = 𝐺 → 𝐺 ⊆ 𝐹)
5		tposss 8170	. . 3 ⊢ (𝐺 ⊆ 𝐹 → tpos 𝐺 ⊆ tpos 𝐹)
6	4, 5	syl 17	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹)
7	3, 6	eqssd 3940	1 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3890  tpos ctpos 8168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-tpos 8169

This theorem is referenced by:  tposeqd  8172  tposeqi  8202