Theorem dmtpos 8240

Description: The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
dmtpos	⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)

Proof of Theorem dmtpos

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5693	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
2		ssel 3928	. . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V)))
3	1, 2	mtoi 202	. . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹)
4		df-rel 5666	. . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V))
5		reldmtpos 8236	. . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
6	3, 4, 5	3imtr4i 295	. . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹)
7		relcnv 6104	. . 3 ⊢ Rel ◡dom 𝐹
8	6, 7	jctir 530	. 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹))
9		vex 3457	. . . . . 6 ⊢ 𝑧 ∈ V
10		brtpos 8237	. . . . . 6 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
11	9, 10	mp1i 14	. . . . 5 ⊢ (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
12	11	exbidv 1954	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧))
13		opex 5443	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
14	13	eldm 5888	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧)
15		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
16		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
17	15, 16	opelcnv 5865	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹)
18		opex 5443	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
19	18	eldm 5888	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)
20	17, 19	bitri 278	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)
21	12, 14, 20	3bitr4g 317	. . 3 ⊢ (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹))
22	21	eqrelrdv2 5779	. 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹)
23	8, 22	mpancom 701	1 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  ⟨cop 4593   class class class wbr 5107   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  Rel wrel 5664  tpos ctpos 8227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-tpos 8228

This theorem is used by:  rntpos  8241  dftpos2  8245  dftpos3  8246  tposfn2  8250