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Theorem tposss 7560
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 5448 . . 3 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
2 dmss 5493 . . . . . 6 (𝐹𝐺 → dom 𝐹 ⊆ dom 𝐺)
3 cnvss 5465 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺dom 𝐹dom 𝐺)
4 unss1 3946 . . . . . 6 (dom 𝐹dom 𝐺 → (dom 𝐹 ∪ {∅}) ⊆ (dom 𝐺 ∪ {∅}))
5 resmpt 5628 . . . . . 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 5599 . . . . 5 ((𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})
86, 7syl6eqssr 3818 . . . 4 (𝐹𝐺 → (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
9 coss2 5449 . . . 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 3773 . 2 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})))
12 df-tpos 7559 . 2 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
13 df-tpos 7559 . 2 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
1411, 12, 133sstr4g 3808 1 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  cun 3732  wss 3734  c0 4081  {csn 4336   cuni 4596  cmpt 4890  ccnv 5278  dom cdm 5279  cres 5281  ccom 5283  tpos ctpos 7558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-mpt 4891  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-res 5291  df-tpos 7559
This theorem is referenced by:  tposeq  7561
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