Theorem ex-rn 28213

Description: Example for df-rn 5560. 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

Step	Hyp	Ref	Expression
1		rneq 5800	. 2 ⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = ran {⟨2, 6⟩, ⟨3, 9⟩})
2		df-pr 4563	. . . 4 ⊢ {⟨2, 6⟩, ⟨3, 9⟩} = ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
3	2	rneqi 5801	. . 3 ⊢ ran {⟨2, 6⟩, ⟨3, 9⟩} = ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩})
4		rnun 5998	. . 3 ⊢ ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩})
5		2nn 11704	. . . . . . 7 ⊢ 2 ∈ ℕ
6	5	elexi 3513	. . . . . 6 ⊢ 2 ∈ V
7	6	rnsnop 6075	. . . . 5 ⊢ ran {⟨2, 6⟩} = {6}
8		3nn 11710	. . . . . . 7 ⊢ 3 ∈ ℕ
9	8	elexi 3513	. . . . . 6 ⊢ 3 ∈ V
10	9	rnsnop 6075	. . . . 5 ⊢ ran {⟨3, 9⟩} = {9}
11	7, 10	uneq12i 4136	. . . 4 ⊢ (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = ({6} ∪ {9})
12		df-pr 4563	. . . 4 ⊢ {6, 9} = ({6} ∪ {9})
13	11, 12	eqtr4i 2847	. . 3 ⊢ (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = {6, 9}
14	3, 4, 13	3eqtri 2848	. 2 ⊢ ran {⟨2, 6⟩, ⟨3, 9⟩} = {6, 9}
15	1, 14	syl6eq 2872	1 ⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})

Syntax hints:   → wi 4   = wceq 1533   ∪ cun 3933  {csn 4560  {cpr 4562  ⟨cop 4566  ran crn 5550  ℕcn 11632  2c2 11686  3c3 11687  6c6 11690  9c9 11693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-1cn 10589

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-nn 11633  df-2 11694  df-3 11695

This theorem is referenced by: (None)