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Theorem tposss 5802
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 4434 . . 3 (𝐹𝐺 → (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x})) ⊆ (𝐺 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x})))
2 dmss 4477 . . . . . 6 (𝐹𝐺 → dom 𝐹 ⊆ dom 𝐺)
3 cnvss 4451 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺dom 𝐹dom 𝐺)
4 unss1 3106 . . . . . 6 (dom 𝐹dom 𝐺 → (dom 𝐹 ∪ {∅}) ⊆ (dom 𝐺 ∪ {∅}))
5 resmpt 4599 . . . . . 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 4578 . . . . 5 ((x (dom 𝐺 ∪ {∅}) ↦ {x}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (x (dom 𝐺 ∪ {∅}) ↦ {x})
86, 7syl6eqssr 2990 . . . 4 (𝐹𝐺 → (x (dom 𝐹 ∪ {∅}) ↦ {x}) ⊆ (x (dom 𝐺 ∪ {∅}) ↦ {x}))
9 coss2 4435 . . . 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 2949 . 2 (𝐹𝐺 → (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x})) ⊆ (𝐺 ∘ (x (dom 𝐺 ∪ {∅}) ↦ {x})))
12 df-tpos 5801 . 2 tpos 𝐹 = (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))
13 df-tpos 5801 . 2 tpos 𝐺 = (𝐺 ∘ (x (dom 𝐺 ∪ {∅}) ↦ {x}))
1411, 12, 133sstr4g 2980 1 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cun 2909  wss 2911  c0 3218  {csn 3367   cuni 3571  cmpt 3809  ccnv 4287  dom cdm 4288  cres 4290  ccom 4292  tpos ctpos 5800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-tpos 5801
This theorem is referenced by:  tposeq  5803
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