Theorem rntpos 8055

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 3436	. . . . 5 ⊢ 𝑧 ∈ V
2	1	elrn 5802	. . . 4 ⊢ (𝑧 ∈ ran tpos 𝐹 ↔ ∃𝑤 𝑤tpos 𝐹𝑧)
3		vex 3436	. . . . . . . . 9 ⊢ 𝑤 ∈ V
4	3, 1	breldm 5817	. . . . . . . 8 ⊢ (𝑤tpos 𝐹𝑧 → 𝑤 ∈ dom tpos 𝐹)
5		dmtpos 8054	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
6	5	eleq2d 2824	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (𝑤 ∈ dom tpos 𝐹 ↔ 𝑤 ∈ ◡dom 𝐹))
7	4, 6	syl5ib 243	. . . . . . 7 ⊢ (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → 𝑤 ∈ ◡dom 𝐹))
8		relcnv 6012	. . . . . . . 8 ⊢ Rel ◡dom 𝐹
9		elrel 5708	. . . . . . . 8 ⊢ ((Rel ◡dom 𝐹 ∧ 𝑤 ∈ ◡dom 𝐹) → ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
10	8, 9	mpan 687	. . . . . . 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 5077	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
13		brtpos 8051	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
14	13	elv 3438	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
15	12, 14	bitrdi 287	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
16		opex 5379	. . . . . . . . 9 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
17	16, 1	brelrn 5851	. . . . . . . 8 ⊢ (⟨𝑦, 𝑥⟩𝐹𝑧 → 𝑧 ∈ ran 𝐹)
18	15, 17	syl6bi 252	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
19	18	exlimivv 1935	. . . . . 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 1936	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑤 𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
22	2, 21	syl5bi 241	. . 3 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹 → 𝑧 ∈ ran 𝐹))
23	1	elrn 5802	. . . 4 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 𝑤𝐹𝑧)
24	3, 1	breldm 5817	. . . . . . 7 ⊢ (𝑤𝐹𝑧 → 𝑤 ∈ dom 𝐹)
25		elrel 5708	. . . . . . . 8 ⊢ ((Rel dom 𝐹 ∧ 𝑤 ∈ dom 𝐹) → ∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩)
26	25	ex 413	. . . . . . 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 5077	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
29	28, 14	bitr4di 289	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
30		opex 5379	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
31	30, 1	brelrn 5851	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹)
32	29, 31	syl6bi 252	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
33	32	exlimivv 1935	. . . . . 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 1936	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑤 𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
36	23, 35	syl5bi 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 2736	1 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   class class class wbr 5074  ^◡ccnv 5588  dom cdm 5589  ran crn 5590  Rel wrel 5594  tpos ctpos 8041

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-tpos 8042

This theorem is referenced by:  tposfo2  8065  oppchofcl  17978  oyoncl  17988