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Mirrors > Home > MPE Home > Th. List > ex-rn | Structured version Visualization version GIF version |
Description: Example for df-rn 5260. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.) |
Ref | Expression |
---|---|
ex-rn | ⊢ (𝐹 = {〈2, 6〉, 〈3, 9〉} → ran 𝐹 = {6, 9}) |
Step | Hyp | Ref | 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〉}) | |
3 | 2 | rneqi 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 | |
6 | 5 | elexi 3365 | . . . . . 6 ⊢ 2 ∈ V |
7 | 6 | rnsnop 5757 | . . . . 5 ⊢ ran {〈2, 6〉} = {6} |
8 | 3nn 11386 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
9 | 8 | elexi 3365 | . . . . . 6 ⊢ 3 ∈ V |
10 | 9 | rnsnop 5757 | . . . . 5 ⊢ ran {〈3, 9〉} = {9} |
11 | 7, 10 | uneq12i 3916 | . . . 4 ⊢ (ran {〈2, 6〉} ∪ ran {〈3, 9〉}) = ({6} ∪ {9}) |
12 | df-pr 4319 | . . . 4 ⊢ {6, 9} = ({6} ∪ {9}) | |
13 | 11, 12 | eqtr4i 2796 | . . 3 ⊢ (ran {〈2, 6〉} ∪ ran {〈3, 9〉}) = {6, 9} |
14 | 3, 4, 13 | 3eqtri 2797 | . 2 ⊢ ran {〈2, 6〉, 〈3, 9〉} = {6, 9} |
15 | 1, 14 | syl6eq 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 ℕcn 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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