Theorem rntpos 8183

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 3445	. . . . 5 ⊢ 𝑧 ∈ V
2	1	elrn 5843	. . . 4 ⊢ (𝑧 ∈ ran tpos 𝐹 ↔ ∃𝑤 𝑤tpos 𝐹𝑧)
3		vex 3445	. . . . . . . . 9 ⊢ 𝑤 ∈ V
4	3, 1	breldm 5858	. . . . . . . 8 ⊢ (𝑤tpos 𝐹𝑧 → 𝑤 ∈ dom tpos 𝐹)
5		dmtpos 8182	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
6	5	eleq2d 2823	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (𝑤 ∈ dom tpos 𝐹 ↔ 𝑤 ∈ ◡dom 𝐹))
7	4, 6	imbitrid 244	. . . . . . 7 ⊢ (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → 𝑤 ∈ ◡dom 𝐹))
8		relcnv 6064	. . . . . . . 8 ⊢ Rel ◡dom 𝐹
9		elrel 5748	. . . . . . . 8 ⊢ ((Rel ◡dom 𝐹 ∧ 𝑤 ∈ ◡dom 𝐹) → ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
10	8, 9	mpan 691	. . . . . . 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 5102	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
13		brtpos 8179	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
14	13	elv 3446	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
15	12, 14	bitrdi 287	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
16		opex 5413	. . . . . . . . 9 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
17	16, 1	brelrn 5892	. . . . . . . 8 ⊢ (⟨𝑦, 𝑥⟩𝐹𝑧 → 𝑧 ∈ ran 𝐹)
18	15, 17	biimtrdi 253	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
19	18	exlimivv 1934	. . . . . 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 1935	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑤 𝑤tpos 𝐹𝑧 → 𝑧 ∈ ran 𝐹))
22	2, 21	biimtrid 242	. . 3 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹 → 𝑧 ∈ ran 𝐹))
23	1	elrn 5843	. . . 4 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 𝑤𝐹𝑧)
24	3, 1	breldm 5858	. . . . . . 7 ⊢ (𝑤𝐹𝑧 → 𝑤 ∈ dom 𝐹)
25		elrel 5748	. . . . . . . 8 ⊢ ((Rel dom 𝐹 ∧ 𝑤 ∈ dom 𝐹) → ∃𝑦∃𝑥 𝑤 = ⟨𝑦, 𝑥⟩)
26	25	ex 412	. . . . . . 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 5102	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
29	28, 14	bitr4di 289	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
30		opex 5413	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
31	30, 1	brelrn 5892	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹)
32	29, 31	biimtrdi 253	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
33	32	exlimivv 1934	. . . . . 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 1935	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑤 𝑤𝐹𝑧 → 𝑧 ∈ ran tpos 𝐹))
36	23, 35	biimtrid 242	. . 3 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran 𝐹 → 𝑧 ∈ ran tpos 𝐹))
37	22, 36	impbid 212	. 2 ⊢ (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹 ↔ 𝑧 ∈ ran 𝐹))
38	37	eqrdv 2735	1 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3441  ⟨cop 4587   class class class wbr 5099  ^◡ccnv 5624  dom cdm 5625  ran crn 5626  Rel wrel 5630  tpos ctpos 8169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501  df-tpos 8170

This theorem is referenced by:  tposfo2  8193  oppchofcl  18187  oyoncl  18197