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Mirrors > Home > MPE Home > Th. List > ex-rn | Structured version Visualization version GIF version |
Description: Example for df-rn 5049. 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 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〉}) | |
3 | 2 | rneqi 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 | |
6 | 5 | elexi 3186 | . . . . . 6 ⊢ 2 ∈ V |
7 | 6 | rnsnop 5534 | . . . . 5 ⊢ ran {〈2, 6〉} = {6} |
8 | 3nn 11063 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
9 | 8 | elexi 3186 | . . . . . 6 ⊢ 3 ∈ V |
10 | 9 | rnsnop 5534 | . . . . 5 ⊢ ran {〈3, 9〉} = {9} |
11 | 7, 10 | uneq12i 3727 | . . . 4 ⊢ (ran {〈2, 6〉} ∪ ran {〈3, 9〉}) = ({6} ∪ {9}) |
12 | df-pr 4128 | . . . 4 ⊢ {6, 9} = ({6} ∪ {9}) | |
13 | 11, 12 | eqtr4i 2635 | . . 3 ⊢ (ran {〈2, 6〉} ∪ ran {〈3, 9〉}) = {6, 9} |
14 | 3, 4, 13 | 3eqtri 2636 | . 2 ⊢ ran {〈2, 6〉, 〈3, 9〉} = {6, 9} |
15 | 1, 14 | syl6eq 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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