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Theorem rntpos 7534
 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.
StepHypRef Expression
1 vex 3343 . . . . 5 𝑧 ∈ V
21elrn 5521 . . . 4 (𝑧 ∈ ran tpos 𝐹 ↔ ∃𝑤 𝑤tpos 𝐹𝑧)
3 vex 3343 . . . . . . . . 9 𝑤 ∈ V
43, 1breldm 5484 . . . . . . . 8 (𝑤tpos 𝐹𝑧𝑤 ∈ dom tpos 𝐹)
5 dmtpos 7533 . . . . . . . . 9 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
65eleq2d 2825 . . . . . . . 8 (Rel dom 𝐹 → (𝑤 ∈ dom tpos 𝐹𝑤dom 𝐹))
74, 6syl5ib 234 . . . . . . 7 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑤dom 𝐹))
8 relcnv 5661 . . . . . . . 8 Rel dom 𝐹
9 elrel 5379 . . . . . . . 8 ((Rel dom 𝐹𝑤dom 𝐹) → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
108, 9mpan 708 . . . . . . 7 (𝑤dom 𝐹 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
117, 10syl6 35 . . . . . 6 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
12 breq1 4807 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
13 brtpos 7530 . . . . . . . . . 10 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
141, 13ax-mp 5 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
1512, 14syl6bb 276 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
16 opex 5081 . . . . . . . . 9 𝑦, 𝑥⟩ ∈ V
1716, 1brelrn 5511 . . . . . . . 8 (⟨𝑦, 𝑥𝐹𝑧𝑧 ∈ ran 𝐹)
1815, 17syl6bi 243 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
1918exlimivv 2009 . . . . . 6 (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2011, 19syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2120exlimdv 2010 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
222, 21syl5bi 232 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
231elrn 5521 . . . 4 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 𝑤𝐹𝑧)
243, 1breldm 5484 . . . . . . 7 (𝑤𝐹𝑧𝑤 ∈ dom 𝐹)
25 elrel 5379 . . . . . . . 8 ((Rel dom 𝐹𝑤 ∈ dom 𝐹) → ∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩)
2625ex 449 . . . . . . 7 (Rel dom 𝐹 → (𝑤 ∈ dom 𝐹 → ∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩))
2724, 26syl5 34 . . . . . 6 (Rel dom 𝐹 → (𝑤𝐹𝑧 → ∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩))
28 breq1 4807 . . . . . . . . 9 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2928, 14syl6bbr 278 . . . . . . . 8 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
30 opex 5081 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
3130, 1brelrn 5511 . . . . . . . 8 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧𝑧 ∈ ran tpos 𝐹)
3229, 31syl6bi 243 . . . . . . 7 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3332exlimivv 2009 . . . . . 6 (∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3427, 33syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3534exlimdv 2010 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3623, 35syl5bi 232 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran 𝐹𝑧 ∈ ran tpos 𝐹))
3722, 36impbid 202 . 2 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
3837eqrdv 2758 1 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632  ∃wex 1853   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327   class class class wbr 4804  ◡ccnv 5265  dom cdm 5266  ran crn 5267  Rel wrel 5271  tpos ctpos 7520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-tpos 7521 This theorem is referenced by:  tposfo2  7544  oppchofcl  17101  oyoncl  17111
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