Theorem tposss 8229

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 5839	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
2		dmss 5890	. . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺)
3		cnvss 5856	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → ◡dom 𝐹 ⊆ ◡dom 𝐺)
4		unss1 4134	. . . . . 6 ⊢ (◡dom 𝐹 ⊆ ◡dom 𝐺 → (◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}))
5		resmpt 6037	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
6	2, 3, 4, 5	4syl 20	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
7		resss 5998	. . . . 5 ⊢ ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})
8	6, 7	eqsstrrdi 3979	. . . 4 ⊢ (𝐹 ⊆ 𝐺 → (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
9		coss2 5840	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
10	8, 9	syl 18	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
11	1, 10	sstrd 3944	. 2 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
12		df-tpos 8228	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
13		df-tpos 8228	. 2 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
14	11, 12, 13	3sstr4g 3987	1 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∪ cun 3900   ⊆ wss 3902  ∅c0 4282  {csn 4587  ∪ cuni 4870   ↦ cmpt 5190  ^◡ccnv 5658  dom cdm 5659   ↾ cres 5661   ∘ ccom 5663  tpos ctpos 8227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-mpt 5191  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-tpos 8228

This theorem is used by:  tposeq  8230