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Theorem ex-rn 28223
 Description: Example for df-rn 5543. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
Assertion
Ref Expression
ex-rn (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})

Proof of Theorem ex-rn
StepHypRef Expression
1 rneq 5783 . 2 (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = ran {⟨2, 6⟩, ⟨3, 9⟩})
2 df-pr 4542 . . . 4 {⟨2, 6⟩, ⟨3, 9⟩} = ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
32rneqi 5784 . . 3 ran {⟨2, 6⟩, ⟨3, 9⟩} = ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
4 rnun 5982 . . 3 ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩})
5 2nn 11698 . . . . . . 7 2 ∈ ℕ
65elexi 3488 . . . . . 6 2 ∈ V
76rnsnop 6059 . . . . 5 ran {⟨2, 6⟩} = {6}
8 3nn 11704 . . . . . . 7 3 ∈ ℕ
98elexi 3488 . . . . . 6 3 ∈ V
109rnsnop 6059 . . . . 5 ran {⟨3, 9⟩} = {9}
117, 10uneq12i 4112 . . . 4 (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = ({6} ∪ {9})
12 df-pr 4542 . . . 4 {6, 9} = ({6} ∪ {9})
1311, 12eqtr4i 2848 . . 3 (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = {6, 9}
143, 4, 133eqtri 2849 . 2 ran {⟨2, 6⟩, ⟨3, 9⟩} = {6, 9}
151, 14syl6eq 2873 1 (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∪ cun 3906  {csn 4539  {cpr 4541  ⟨cop 4545  ran crn 5533  ℕcn 11625  2c2 11680  3c3 11681  6c6 11684  9c9 11687 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-1cn 10584 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-nn 11626  df-2 11688  df-3 11689 This theorem is referenced by: (None)
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