Theorem tposss 7876

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 5690	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
2		dmss 5735	. . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺)
3		cnvss 5707	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → ◡dom 𝐹 ⊆ ◡dom 𝐺)
4		unss1 4106	. . . . . 6 ⊢ (◡dom 𝐹 ⊆ ◡dom 𝐺 → (◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}))
5		resmpt 5872	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ⊆ (◡dom 𝐺 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
6	2, 3, 4, 5	4syl 19	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
7		resss 5843	. . . . 5 ⊢ ((𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})
8	6, 7	eqsstrrdi 3970	. . . 4 ⊢ (𝐹 ⊆ 𝐺 → (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
9		coss2 5691	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}) → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
10	8, 9	syl 17	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (𝐺 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
11	1, 10	sstrd 3925	. 2 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥})))
12		df-tpos 7875	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
13		df-tpos 7875	. 2 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ (◡dom 𝐺 ∪ {∅}) ↦ ∪ ◡{𝑥}))
14	11, 12, 13	3sstr4g 3960	1 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1538   ∪ cun 3879   ⊆ wss 3881  ∅c0 4243  {csn 4525  ∪ cuni 4800   ↦ cmpt 5110  ^◡ccnv 5518  dom cdm 5519   ↾ cres 5521   ∘ ccom 5523  tpos ctpos 7874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-tpos 7875

This theorem is referenced by:  tposeq  7877