Theorem tposeq 8175

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 8174	. . 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 8174	. . 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 1547   ⊆ wss 3890  tpos ctpos 8172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-mpt 5161  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-res 5637  df-tpos 8173

This theorem is referenced by:  tposeqd  8176  tposeqi  8206