Theorem dmtpos 8279

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 5734	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
2		ssel 4002	. . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V)))
3	1, 2	mtoi 199	. . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹)
4		df-rel 5707	. . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V))
5		reldmtpos 8275	. . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
6	3, 4, 5	3imtr4i 292	. . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹)
7		relcnv 6134	. . 3 ⊢ Rel ◡dom 𝐹
8	6, 7	jctir 520	. 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹))
9		vex 3492	. . . . . 6 ⊢ 𝑧 ∈ V
10		brtpos 8276	. . . . . 6 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
11	9, 10	mp1i 13	. . . . 5 ⊢ (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
12	11	exbidv 1920	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧))
13		opex 5484	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
14	13	eldm 5925	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧)
15		vex 3492	. . . . . 6 ⊢ 𝑥 ∈ V
16		vex 3492	. . . . . 6 ⊢ 𝑦 ∈ V
17	15, 16	opelcnv 5906	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹)
18		opex 5484	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
19	18	eldm 5925	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)
20	17, 19	bitri 275	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)
21	12, 14, 20	3bitr4g 314	. . 3 ⊢ (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹))
22	21	eqrelrdv2 5819	. 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹)
23	8, 22	mpancom 687	1 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ⟨cop 4654   class class class wbr 5166   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  Rel wrel 5705  tpos ctpos 8266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-tpos 8267

This theorem is referenced by:  rntpos  8280  dftpos2  8284  dftpos3  8285  tposfn2  8289