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Theorem tposss 5771
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 4406 . . 3 (𝐹𝐺 → (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x})) ⊆ (𝐺 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x})))
2 dmss 4449 . . . . . 6 (𝐹𝐺 → dom 𝐹 ⊆ dom 𝐺)
3 cnvss 4423 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺dom 𝐹dom 𝐺)
4 unss1 3080 . . . . . 6 (dom 𝐹dom 𝐺 → (dom 𝐹 ∪ {∅}) ⊆ (dom 𝐺 ∪ {∅}))
5 resmpt 4571 . . . . . 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 4550 . . . . 5 ((x (dom 𝐺 ∪ {∅}) ↦ {x}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (x (dom 𝐺 ∪ {∅}) ↦ {x})
86, 7syl6eqssr 2964 . . . 4 (𝐹𝐺 → (x (dom 𝐹 ∪ {∅}) ↦ {x}) ⊆ (x (dom 𝐺 ∪ {∅}) ↦ {x}))
9 coss2 4407 . . . 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 2923 . 2 (𝐹𝐺 → (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x})) ⊆ (𝐺 ∘ (x (dom 𝐺 ∪ {∅}) ↦ {x})))
12 df-tpos 5770 . 2 tpos 𝐹 = (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))
13 df-tpos 5770 . 2 tpos 𝐺 = (𝐺 ∘ (x (dom 𝐺 ∪ {∅}) ↦ {x}))
1411, 12, 133sstr4g 2954 1 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1223  cun 2883  wss 2885  c0 3192  {csn 3339   cuni 3543  cmpt 3781  ccnv 4259  dom cdm 4260  cres 4262  ccom 4264  tpos ctpos 5769
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-mpt 3783  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-res 4272  df-tpos 5770
This theorem is referenced by:  tposeq  5772
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