Theorem tposss 6313

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 4822	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
2		dmss 4866	. . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺)
3		cnvss 4840	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → ◡dom 𝐹 ⊆ ◡dom 𝐺)
4		unss1 3333	. . . . . 6 ⊢ (◡dom 𝐹 ⊆ ◡dom 𝐺 → (◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}))
5		resmpt 4995	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
6	2, 3, 4, 5	4syl 18	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
7		resss 4971	. . . . 5 ⊢ ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})
8	6, 7	eqsstrrdi 3237	. . . 4 ⊢ (𝐹 ⊆ 𝐺 → (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
9		coss2 4823	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
10	8, 9	syl 14	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
11	1, 10	sstrd 3194	. 2 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
12		df-tpos 6312	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
13		df-tpos 6312	. 2 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
14	11, 12, 13	3sstr4g 3227	1 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1364   ∪ cun 3155   ⊆ wss 3157  ∅c0 3451  {csn 3623  ∪ cuni 3840   ↦ cmpt 4095  ^◡ccnv 4663  dom cdm 4664   ↾ cres 4666   ∘ ccom 4668  tpos ctpos 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-mpt 4097  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-res 4676  df-tpos 6312

This theorem is referenced by:  tposeq  6314