Theorem tposssxp 6225

Description: The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)

Assertion

Ref	Expression
tposssxp	⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹)

Proof of Theorem tposssxp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tpos 6221	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2		cossxp 5131	. . 3 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹)
3	1, 2	eqsstri 3179	. 2 ⊢ tpos 𝐹 ⊆ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹)
4		eqid 2170	. . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
5	4	dmmptss 5105	. . 3 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅})
6		xpss1 4719	. . 3 ⊢ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅}) → (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹))
7	5, 6	ax-mp 5	. 2 ⊢ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹)
8	3, 7	sstri 3156	1 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹)

Syntax hints:   ∪ cun 3119   ⊆ wss 3121  ∅c0 3414  {csn 3581  ∪ cuni 3794   ↦ cmpt 4048   × cxp 4607  ^◡ccnv 4608  dom cdm 4609  ran crn 4610   ∘ ccom 4613  tpos ctpos 6220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-mpt 4050  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-tpos 6221

This theorem is referenced by:  reltpos  6226  tposexg  6234