Theorem dmtpos 8025

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 5614	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
2		ssel 3910	. . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V)))
3	1, 2	mtoi 198	. . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹)
4		df-rel 5587	. . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V))
5		reldmtpos 8021	. . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
6	3, 4, 5	3imtr4i 291	. . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹)
7		relcnv 6001	. . 3 ⊢ Rel ◡dom 𝐹
8	6, 7	jctir 520	. 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹))
9		vex 3426	. . . . . 6 ⊢ 𝑧 ∈ V
10		brtpos 8022	. . . . . 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 1925	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧))
13		opex 5373	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
14	13	eldm 5798	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧)
15		vex 3426	. . . . . 6 ⊢ 𝑥 ∈ V
16		vex 3426	. . . . . 6 ⊢ 𝑦 ∈ V
17	15, 16	opelcnv 5779	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹)
18		opex 5373	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
19	18	eldm 5798	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)
20	17, 19	bitri 274	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)
21	12, 14, 20	3bitr4g 313	. . 3 ⊢ (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹))
22	21	eqrelrdv2 5694	. 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹)
23	8, 22	mpancom 684	1 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ⟨cop 4564   class class class wbr 5070   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  Rel wrel 5585  tpos ctpos 8012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-tpos 8013

This theorem is referenced by:  rntpos  8026  dftpos2  8030  dftpos3  8031  tposfn2  8035