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Theorem tposss 8239
Description: Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
tposss (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Proof of Theorem tposss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 coss1 5862 . . 3 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
2 dmss 5909 . . . . . 6 (𝐹𝐺 → dom 𝐹 ⊆ dom 𝐺)
3 cnvss 5879 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺dom 𝐹dom 𝐺)
4 unss1 4181 . . . . . 6 (dom 𝐹dom 𝐺 → (dom 𝐹 ∪ {∅}) ⊆ (dom 𝐺 ∪ {∅}))
5 resmpt 6046 . . . . . 6 ((dom 𝐹 ∪ {∅}) ⊆ (dom 𝐺 ∪ {∅}) → ((𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
62, 3, 4, 54syl 19 . . . . 5 (𝐹𝐺 → ((𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
7 resss 6011 . . . . 5 ((𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})
86, 7eqsstrrdi 4037 . . . 4 (𝐹𝐺 → (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
9 coss2 5863 . . . 4 ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) → (𝐺 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})))
108, 9syl 17 . . 3 (𝐹𝐺 → (𝐺 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})))
111, 10sstrd 3992 . 2 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})))
12 df-tpos 8238 . 2 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
13 df-tpos 8238 . 2 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
1411, 12, 133sstr4g 4027 1 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cun 3947  wss 3949  c0 4326  {csn 4632   cuni 4912  cmpt 5235  ccnv 5681  dom cdm 5682  cres 5684  ccom 5686  tpos ctpos 8237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-mpt 5236  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-tpos 8238
This theorem is referenced by:  tposeq  8240
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