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Theorem ex-rn 27168
Description: Example for df-rn 5090. 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 5316 . 2 (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = ran {⟨2, 6⟩, ⟨3, 9⟩})
2 df-pr 4156 . . . 4 {⟨2, 6⟩, ⟨3, 9⟩} = ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
32rneqi 5317 . . 3 ran {⟨2, 6⟩, ⟨3, 9⟩} = ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
4 rnun 5505 . . 3 ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩})
5 2nn 11137 . . . . . . 7 2 ∈ ℕ
65elexi 3202 . . . . . 6 2 ∈ V
76rnsnop 5580 . . . . 5 ran {⟨2, 6⟩} = {6}
8 3nn 11138 . . . . . . 7 3 ∈ ℕ
98elexi 3202 . . . . . 6 3 ∈ V
109rnsnop 5580 . . . . 5 ran {⟨3, 9⟩} = {9}
117, 10uneq12i 3748 . . . 4 (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = ({6} ∪ {9})
12 df-pr 4156 . . . 4 {6, 9} = ({6} ∪ {9})
1311, 12eqtr4i 2646 . . 3 (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = {6, 9}
143, 4, 133eqtri 2647 . 2 ran {⟨2, 6⟩, ⟨3, 9⟩} = {6, 9}
151, 14syl6eq 2671 1 (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  cun 3557  {csn 4153  {cpr 4155  cop 4159  ran crn 5080  cn 10972  2c2 11022  3c3 11023  6c6 11026  9c9 11029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-1cn 9946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-nn 10973  df-2 11031  df-3 11032
This theorem is referenced by: (None)
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