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Theorem ex-rn 26689
 Description: Example for df-rn 5049. 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 5272 . 2 (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = ran {⟨2, 6⟩, ⟨3, 9⟩})
2 df-pr 4128 . . . 4 {⟨2, 6⟩, ⟨3, 9⟩} = ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
32rneqi 5273 . . 3 ran {⟨2, 6⟩, ⟨3, 9⟩} = ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
4 rnun 5460 . . 3 ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩})
5 2nn 11062 . . . . . . 7 2 ∈ ℕ
65elexi 3186 . . . . . 6 2 ∈ V
76rnsnop 5534 . . . . 5 ran {⟨2, 6⟩} = {6}
8 3nn 11063 . . . . . . 7 3 ∈ ℕ
98elexi 3186 . . . . . 6 3 ∈ V
109rnsnop 5534 . . . . 5 ran {⟨3, 9⟩} = {9}
117, 10uneq12i 3727 . . . 4 (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = ({6} ∪ {9})
12 df-pr 4128 . . . 4 {6, 9} = ({6} ∪ {9})
1311, 12eqtr4i 2635 . . 3 (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = {6, 9}
143, 4, 133eqtri 2636 . 2 ran {⟨2, 6⟩, ⟨3, 9⟩} = {6, 9}
151, 14syl6eq 2660 1 (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∪ cun 3538  {csn 4125  {cpr 4127  ⟨cop 4131  ran crn 5039  ℕcn 10897  2c2 10947  3c3 10948  6c6 10951  9c9 10954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-1cn 9873 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-2 10956  df-3 10957 This theorem is referenced by: (None)
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