Theorem tposeq 8207

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 4005	. . 3 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺)
2		tposss 8206	. . 3 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
3	1, 2	syl 17	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
4		eqimss2 4006	. . 3 ⊢ (𝐹 = 𝐺 → 𝐺 ⊆ 𝐹)
5		tposss 8206	. . 3 ⊢ (𝐺 ⊆ 𝐹 → tpos 𝐺 ⊆ tpos 𝐹)
6	4, 5	syl 17	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹)
7	3, 6	eqssd 3964	1 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3914  tpos ctpos 8204

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-mpt 5189  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-tpos 8205

This theorem is referenced by:  tposeqd  8208  tposeqi  8238