Theorem tposss 5861

Description: Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
tposss	⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Proof of Theorem tposss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coss1 4491	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
2		dmss 4534	. . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺)
3		cnvss 4508	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → ◡dom 𝐹 ⊆ ◡dom 𝐺)
4		unss1 3112	. . . . . 6 ⊢ (◡dom 𝐹 ⊆ ◡dom 𝐺 → (◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}))
5		resmpt 4656	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
6	2, 3, 4, 5	4syl 18	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
7		resss 4635	. . . . 5 ⊢ ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})
8	6, 7	syl6eqssr 2996	. . . 4 ⊢ (𝐹 ⊆ 𝐺 → (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
9		coss2 4492	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
10	8, 9	syl 14	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
11	1, 10	sstrd 2955	. 2 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
12		df-tpos 5860	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
13		df-tpos 5860	. 2 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
14	11, 12, 13	3sstr4g 2986	1 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1243   ∪ cun 2915   ⊆ wss 2917  ∅c0 3224  {csn 3375  ∪ cuni 3580   ↦ cmpt 3818  ^◡ccnv 4344  dom cdm 4345   ↾ cres 4347   ∘ ccom 4349  tpos ctpos 5859

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-tpos 5860

This theorem is referenced by:  tposeq  5862