Theorem tposeq 8178

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 3980	. . 3 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺)
2		tposss 8177	. . 3 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
3	1, 2	syl 17	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
4		eqimss2 3981	. . 3 ⊢ (𝐹 = 𝐺 → 𝐺 ⊆ 𝐹)
5		tposss 8177	. . 3 ⊢ (𝐺 ⊆ 𝐹 → tpos 𝐺 ⊆ tpos 𝐹)
6	4, 5	syl 17	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹)
7	3, 6	eqssd 3939	1 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3889  tpos ctpos 8175

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-tpos 8176

This theorem is referenced by:  tposeqd  8179  tposeqi  8209