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Theorem ex-rn 27632
Description: Example for df-rn 5260. 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 5487 . 2 (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = ran {⟨2, 6⟩, ⟨3, 9⟩})
2 df-pr 4319 . . . 4 {⟨2, 6⟩, ⟨3, 9⟩} = ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
32rneqi 5488 . . 3 ran {⟨2, 6⟩, ⟨3, 9⟩} = ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
4 rnun 5680 . . 3 ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩})
5 2nn 11385 . . . . . . 7 2 ∈ ℕ
65elexi 3365 . . . . . 6 2 ∈ V
76rnsnop 5757 . . . . 5 ran {⟨2, 6⟩} = {6}
8 3nn 11386 . . . . . . 7 3 ∈ ℕ
98elexi 3365 . . . . . 6 3 ∈ V
109rnsnop 5757 . . . . 5 ran {⟨3, 9⟩} = {9}
117, 10uneq12i 3916 . . . 4 (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = ({6} ∪ {9})
12 df-pr 4319 . . . 4 {6, 9} = ({6} ∪ {9})
1311, 12eqtr4i 2796 . . 3 (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = {6, 9}
143, 4, 133eqtri 2797 . 2 ran {⟨2, 6⟩, ⟨3, 9⟩} = {6, 9}
151, 14syl6eq 2821 1 (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  cun 3721  {csn 4316  {cpr 4318  cop 4322  ran crn 5250  cn 11220  2c2 11270  3c3 11271  6c6 11274  9c9 11277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-1cn 10194
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6794  df-om 7211  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-nn 11221  df-2 11279  df-3 11280
This theorem is referenced by: (None)
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