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Mirrors > Home > MPE Home > Th. List > ex-rn | Structured version Visualization version GIF version |
Description: Example for df-rn 5695. 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 5944 | . 2 ⊢ (𝐹 = {〈2, 6〉, 〈3, 9〉} → ran 𝐹 = ran {〈2, 6〉, 〈3, 9〉}) | |
2 | df-pr 4636 | . . . 4 ⊢ {〈2, 6〉, 〈3, 9〉} = ({〈2, 6〉} ∪ {〈3, 9〉}) | |
3 | 2 | rneqi 5945 | . . 3 ⊢ ran {〈2, 6〉, 〈3, 9〉} = ran ({〈2, 6〉} ∪ {〈3, 9〉}) |
4 | rnun 6159 | . . 3 ⊢ ran ({〈2, 6〉} ∪ {〈3, 9〉}) = (ran {〈2, 6〉} ∪ ran {〈3, 9〉}) | |
5 | 2nn 12339 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
6 | 5 | elexi 3484 | . . . . . 6 ⊢ 2 ∈ V |
7 | 6 | rnsnop 6237 | . . . . 5 ⊢ ran {〈2, 6〉} = {6} |
8 | 3nn 12345 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
9 | 8 | elexi 3484 | . . . . . 6 ⊢ 3 ∈ V |
10 | 9 | rnsnop 6237 | . . . . 5 ⊢ ran {〈3, 9〉} = {9} |
11 | 7, 10 | uneq12i 4161 | . . . 4 ⊢ (ran {〈2, 6〉} ∪ ran {〈3, 9〉}) = ({6} ∪ {9}) |
12 | df-pr 4636 | . . . 4 ⊢ {6, 9} = ({6} ∪ {9}) | |
13 | 11, 12 | eqtr4i 2757 | . . 3 ⊢ (ran {〈2, 6〉} ∪ ran {〈3, 9〉}) = {6, 9} |
14 | 3, 4, 13 | 3eqtri 2758 | . 2 ⊢ ran {〈2, 6〉, 〈3, 9〉} = {6, 9} |
15 | 1, 14 | eqtrdi 2782 | 1 ⊢ (𝐹 = {〈2, 6〉, 〈3, 9〉} → ran 𝐹 = {6, 9}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∪ cun 3945 {csn 4633 {cpr 4635 〈cop 4639 ran crn 5685 ℕcn 12266 2c2 12321 3c3 12322 6c6 12325 9c9 12328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 ax-un 7748 ax-1cn 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-ov 7429 df-om 7879 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-nn 12267 df-2 12329 df-3 12330 |
This theorem is referenced by: (None) |
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