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Mirrors > Home > MPE Home > Th. List > ex-cnv | Structured version Visualization version GIF version |
Description: Example for df-cnv 5527. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
ex-cnv | ⊢ ◡{〈2, 6〉, 〈3, 9〉} = {〈6, 2〉, 〈9, 3〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvun 5968 | . . 3 ⊢ ◡({〈2, 6〉} ∪ {〈3, 9〉}) = (◡{〈2, 6〉} ∪ ◡{〈3, 9〉}) | |
2 | 2nn 11698 | . . . . . 6 ⊢ 2 ∈ ℕ | |
3 | 2 | elexi 3460 | . . . . 5 ⊢ 2 ∈ V |
4 | 6nn 11714 | . . . . . 6 ⊢ 6 ∈ ℕ | |
5 | 4 | elexi 3460 | . . . . 5 ⊢ 6 ∈ V |
6 | 3, 5 | cnvsn 6050 | . . . 4 ⊢ ◡{〈2, 6〉} = {〈6, 2〉} |
7 | 3nn 11704 | . . . . . 6 ⊢ 3 ∈ ℕ | |
8 | 7 | elexi 3460 | . . . . 5 ⊢ 3 ∈ V |
9 | 9nn 11723 | . . . . . 6 ⊢ 9 ∈ ℕ | |
10 | 9 | elexi 3460 | . . . . 5 ⊢ 9 ∈ V |
11 | 8, 10 | cnvsn 6050 | . . . 4 ⊢ ◡{〈3, 9〉} = {〈9, 3〉} |
12 | 6, 11 | uneq12i 4088 | . . 3 ⊢ (◡{〈2, 6〉} ∪ ◡{〈3, 9〉}) = ({〈6, 2〉} ∪ {〈9, 3〉}) |
13 | 1, 12 | eqtri 2821 | . 2 ⊢ ◡({〈2, 6〉} ∪ {〈3, 9〉}) = ({〈6, 2〉} ∪ {〈9, 3〉}) |
14 | df-pr 4528 | . . 3 ⊢ {〈2, 6〉, 〈3, 9〉} = ({〈2, 6〉} ∪ {〈3, 9〉}) | |
15 | 14 | cnveqi 5709 | . 2 ⊢ ◡{〈2, 6〉, 〈3, 9〉} = ◡({〈2, 6〉} ∪ {〈3, 9〉}) |
16 | df-pr 4528 | . 2 ⊢ {〈6, 2〉, 〈9, 3〉} = ({〈6, 2〉} ∪ {〈9, 3〉}) | |
17 | 13, 15, 16 | 3eqtr4i 2831 | 1 ⊢ ◡{〈2, 6〉, 〈3, 9〉} = {〈6, 2〉, 〈9, 3〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∪ cun 3879 {csn 4525 {cpr 4527 〈cop 4531 ◡ccnv 5518 ℕcn 11625 2c2 11680 3c3 11681 6c6 11684 9c9 11687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 |
This theorem is referenced by: (None) |
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