Theorem rntpos 8224

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

Assertion

Ref	Expression
rntpos	⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Proof of Theorem rntpos

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

Step	Hyp	Ref	Expression
1		vex 3479	. . . . 5 ⊢ 𝑧 ∈ V
2	1	elrn 5894	. . . 4 ⊢ (𝑧 ∈ ran tpos 𝐹 ↔ ∃𝑤 𝑤tpos 𝐹𝑧)
3		vex 3479	. . . . . . . . 9 ⊢ 𝑤 ∈ V
4	3, 1	breldm 5909	. . . . . . . 8 ⊢ (𝑤tpos 𝐹𝑧 → 𝑤 ∈ dom tpos 𝐹)
5		dmtpos 8223	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
6	5	eleq2d 2820	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (𝑤 ∈ dom tpos 𝐹 ↔ 𝑤 ∈ ◡dom 𝐹))
7	4, 6	imbitrid 243	. . . . . . 7 ⊢ (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → 𝑤 ∈ ◡dom 𝐹))
8		relcnv 6104	. . . . . . . 8 ⊢ Rel ◡dom 𝐹
9		elrel 5799	. . . . . . . 8 ⊢ ((Rel ◡dom 𝐹 ∧ 𝑤 ∈ ◡dom 𝐹) → ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
10	8, 9	mpan 689	. . . . . . 7 ⊢ (𝑤 ∈ ◡dom 𝐹 → ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
11	7, 10	syl6 35	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
12		breq1 5152	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
13		brtpos 8220	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
14	13	elv 3481	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
15	12, 14	bitrdi 287	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
16		opex 5465	. . . . . . . . 9 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
17	16, 1	brelrn 5942	. . . . . . . 8 ⊢ (⟨𝑦, 𝑥⟩𝐹𝑧 → 𝑧 ∈ ran 𝐹)
18	15, 17	syl6bi 253	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
19	18	exlimivv 1936	. . . . . 6 ⊢ (∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
20	11, 19	syli 39	. . . . 5 ⊢ (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
21	20	exlimdv 1937	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑤 𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
22	2, 21	biimtrid 241	. . 3 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹 → 𝑧 ∈ ran 𝐹))
23	1	elrn 5894	. . . 4 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 𝑤𝐹𝑧)
24	3, 1	breldm 5909	. . . . . . 7 ⊢ (𝑤𝐹𝑧 → 𝑤 ∈ dom 𝐹)
25		elrel 5799	. . . . . . . 8 ⊢ ((Rel dom 𝐹 ∧ 𝑤 ∈ dom 𝐹) → ∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩)
26	25	ex 414	. . . . . . 7 ⊢ (Rel dom 𝐹 → (𝑤 ∈ dom 𝐹 → ∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩))
27	24, 26	syl5 34	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑤𝐹𝑧 → ∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩))
28		breq1 5152	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
29	28, 14	bitr4di 289	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
30		opex 5465	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
31	30, 1	brelrn 5942	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹)
32	29, 31	syl6bi 253	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
33	32	exlimivv 1936	. . . . . 6 ⊢ (∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
34	27, 33	syli 39	. . . . 5 ⊢ (Rel dom 𝐹 → (𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
35	34	exlimdv 1937	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑤 𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
36	23, 35	biimtrid 241	. . 3 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran 𝐹 → 𝑧 ∈ ran tpos 𝐹))
37	22, 36	impbid 211	. 2 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹 ↔ 𝑧 ∈ ran 𝐹))
38	37	eqrdv 2731	1 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475  ⟨cop 4635   class class class wbr 5149  ^◡ccnv 5676  dom cdm 5677  ran crn 5678  Rel wrel 5682  tpos ctpos 8210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-tpos 8211

This theorem is referenced by:  tposfo2  8234  oppchofcl  18213  oyoncl  18223