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Theorem tposss 6240
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 4777 . . 3 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
2 dmss 4821 . . . . . 6 (𝐹𝐺 → dom 𝐹 ⊆ dom 𝐺)
3 cnvss 4795 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺dom 𝐹dom 𝐺)
4 unss1 3304 . . . . . 6 (dom 𝐹dom 𝐺 → (dom 𝐹 ∪ {∅}) ⊆ (dom 𝐺 ∪ {∅}))
5 resmpt 4950 . . . . . 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 4926 . . . . 5 ((𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})
86, 7eqsstrrdi 3208 . . . 4 (𝐹𝐺 → (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
9 coss2 4778 . . . 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 3165 . 2 (𝐹𝐺 → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥})))
12 df-tpos 6239 . 2 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
13 df-tpos 6239 . 2 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (dom 𝐺 ∪ {∅}) ↦ {𝑥}))
1411, 12, 133sstr4g 3198 1 (𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  cun 3127  wss 3129  c0 3422  {csn 3591   cuni 3807  cmpt 4061  ccnv 4621  dom cdm 4622  cres 4624  ccom 4626  tpos ctpos 6238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-mpt 4063  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-res 4634  df-tpos 6239
This theorem is referenced by:  tposeq  6241
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