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Theorem tposss 5892
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 4519 . . 3 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
2 dmss 4562 . . . . . 6 (𝐹𝐺 → dom 𝐹 ⊆ dom 𝐺)
3 cnvss 4536 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺dom 𝐹dom 𝐺)
4 unss1 3140 . . . . . 6 (dom 𝐹dom 𝐺 → (dom 𝐹 ∪ {∅}) ⊆ (dom 𝐺 ∪ {∅}))
5 resmpt 4684 . . . . . 6 ((dom 𝐹 ∪ {∅}) ⊆ (dom 𝐺 ∪ {∅}) → ((𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
62, 3, 4, 54syl 18 . . . . 5 (𝐹𝐺 → ((𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
7 resss 4663 . . . . 5 ((𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})
86, 7syl6eqssr 3024 . . . 4 (𝐹𝐺 → (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
9 coss2 4520 . . . 4 ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) → (𝐺 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})))
108, 9syl 14 . . 3 (𝐹𝐺 → (𝐺 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})))
111, 10sstrd 2983 . 2 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})))
12 df-tpos 5891 . 2 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
13 df-tpos 5891 . 2 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
1411, 12, 133sstr4g 3014 1 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  cun 2943  wss 2945  c0 3252  {csn 3403   cuni 3608  cmpt 3846  ccnv 4372  dom cdm 4373  cres 4375  ccom 4377  tpos ctpos 5890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-res 4385  df-tpos 5891
This theorem is referenced by:  tposeq  5893
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