Theorem rntpos 8223

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 3461	. . . . 5 ⊢ 𝑧 ∈ V
2	1	elrn 5874	. . . 4 ⊢ (𝑧 ∈ ran tpos 𝐹 ↔ ∃𝑤 𝑤tpos 𝐹𝑧)
3		vex 3461	. . . . . . . . 9 ⊢ 𝑤 ∈ V
4	3, 1	breldm 5889	. . . . . . . 8 ⊢ (𝑤tpos 𝐹𝑧 → 𝑤 ∈ dom tpos 𝐹)
5		dmtpos 8222	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
6	5	eleq2d 2851	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (𝑤 ∈ dom tpos 𝐹 ↔ 𝑤 ∈ ◡dom 𝐹))
7	4, 6	imbitrid 247	. . . . . . 7 ⊢ (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → 𝑤 ∈ ◡dom 𝐹))
8		relcnv 6097	. . . . . . . 8 ⊢ Rel ◡dom 𝐹
9		elrel 5775	. . . . . . . 8 ⊢ ((Rel ◡dom 𝐹 ∧ 𝑤 ∈ ◡dom 𝐹) → ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
10	8, 9	mpan 702	. . . . . . 7 ⊢ (𝑤 ∈ ◡dom 𝐹 → ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
11	7, 10	syl6 36	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
12		breq1 5108	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
13		brtpos 8219	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
14	13	elv 3462	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
15	12, 14	bitrdi 290	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
16		opex 5436	. . . . . . . . 9 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
17	16, 1	brelrn 5923	. . . . . . . 8 ⊢ (⟨𝑦, 𝑥⟩𝐹𝑧 → 𝑧 ∈ ran 𝐹)
18	15, 17	biimtrdi 256	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
19	18	exlimivv 1955	. . . . . 6 ⊢ (∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
20	11, 19	syli 40	. . . . 5 ⊢ (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
21	20	exlimdv 1956	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑤 𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
22	2, 21	biimtrid 245	. . 3 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹 → 𝑧 ∈ ran 𝐹))
23	1	elrn 5874	. . . 4 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 𝑤𝐹𝑧)
24	3, 1	breldm 5889	. . . . . . 7 ⊢ (𝑤𝐹𝑧 → 𝑤 ∈ dom 𝐹)
25		elrel 5775	. . . . . . . 8 ⊢ ((Rel dom 𝐹 ∧ 𝑤 ∈ dom 𝐹) → ∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩)
26	25	ex 417	. . . . . . 7 ⊢ (Rel dom 𝐹 → (𝑤 ∈ dom 𝐹 → ∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩))
27	24, 26	syl5 35	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑤𝐹𝑧 → ∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩))
28		breq1 5108	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
29	28, 14	bitr4di 292	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
30		opex 5436	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
31	30, 1	brelrn 5923	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹)
32	29, 31	biimtrdi 256	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
33	32	exlimivv 1955	. . . . . 6 ⊢ (∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
34	27, 33	syli 40	. . . . 5 ⊢ (Rel dom 𝐹 → (𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
35	34	exlimdv 1956	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑤 𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
36	23, 35	biimtrid 245	. . 3 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran 𝐹 → 𝑧 ∈ ran tpos 𝐹))
37	22, 36	impbid 215	. 2 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹 ↔ 𝑧 ∈ ran 𝐹))
38	37	eqrdv 2763	1 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563  ∃wex 1802   ∈ wcel 2145  Vcvv 3457  ⟨cop 4591   class class class wbr 5105  ^◡ccnv 5651  dom cdm 5652  ran crn 5653  Rel wrel 5657  tpos ctpos 8209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533  df-tpos 8210

This theorem is referenced by:  tposfo2  8233  oppchofcl  18306  oyoncl  18316