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Theorem tposss 8162
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 5815 . . 3 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
2 dmss 5862 . . . . . 6 (𝐹𝐺 → dom 𝐹 ⊆ dom 𝐺)
3 cnvss 5832 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺dom 𝐹dom 𝐺)
4 unss1 4143 . . . . . 6 (dom 𝐹dom 𝐺 → (dom 𝐹 ∪ {∅}) ⊆ (dom 𝐺 ∪ {∅}))
5 resmpt 5995 . . . . . 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 5966 . . . . 5 ((𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})
86, 7eqsstrrdi 4003 . . . 4 (𝐹𝐺 → (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
9 coss2 5816 . . . 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 3958 . 2 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})))
12 df-tpos 8161 . 2 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
13 df-tpos 8161 . 2 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
1411, 12, 133sstr4g 3993 1 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cun 3912  wss 3914  c0 4286  {csn 4590   cuni 4869  cmpt 5192  ccnv 5636  dom cdm 5637  cres 5639  ccom 5641  tpos ctpos 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-mpt 5193  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-res 5649  df-tpos 8161
This theorem is referenced by:  tposeq  8163
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