Theorem tposss 8253

Description: Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
tposss	⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Proof of Theorem tposss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coss1 5865	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
2		dmss 5912	. . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺)
3		cnvss 5882	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → ◡dom 𝐹 ⊆ ◡dom 𝐺)
4		unss1 4184	. . . . . 6 ⊢ (◡dom 𝐹 ⊆ ◡dom 𝐺 → (◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}))
5		resmpt 6054	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
6	2, 3, 4, 5	4syl 19	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
7		resss 6018	. . . . 5 ⊢ ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})
8	6, 7	eqsstrrdi 4028	. . . 4 ⊢ (𝐹 ⊆ 𝐺 → (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
9		coss2 5866	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
10	8, 9	syl 17	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
11	1, 10	sstrd 3993	. 2 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
12		df-tpos 8252	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
13		df-tpos 8252	. 2 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
14	11, 12, 13	3sstr4g 4036	1 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1539   ∪ cun 3948   ⊆ wss 3950  ∅c0 4332  {csn 4625  ∪ cuni 4906   ↦ cmpt 5224  ^◡ccnv 5683  dom cdm 5684   ↾ cres 5686   ∘ ccom 5688  tpos ctpos 8251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-tpos 8252

This theorem is referenced by:  tposeq  8254