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Mirrors > Home > MPE Home > Th. List > ex-cnv | Structured version Visualization version GIF version |
Description: Example for df-cnv 5682. 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 6146 | . . 3 ⊢ ◡({〈2, 6〉} ∪ {〈3, 9〉}) = (◡{〈2, 6〉} ∪ ◡{〈3, 9〉}) | |
2 | 2nn 12331 | . . . . . 6 ⊢ 2 ∈ ℕ | |
3 | 2 | elexi 3484 | . . . . 5 ⊢ 2 ∈ V |
4 | 6nn 12347 | . . . . . 6 ⊢ 6 ∈ ℕ | |
5 | 4 | elexi 3484 | . . . . 5 ⊢ 6 ∈ V |
6 | 3, 5 | cnvsn 6229 | . . . 4 ⊢ ◡{〈2, 6〉} = {〈6, 2〉} |
7 | 3nn 12337 | . . . . . 6 ⊢ 3 ∈ ℕ | |
8 | 7 | elexi 3484 | . . . . 5 ⊢ 3 ∈ V |
9 | 9nn 12356 | . . . . . 6 ⊢ 9 ∈ ℕ | |
10 | 9 | elexi 3484 | . . . . 5 ⊢ 9 ∈ V |
11 | 8, 10 | cnvsn 6229 | . . . 4 ⊢ ◡{〈3, 9〉} = {〈9, 3〉} |
12 | 6, 11 | uneq12i 4158 | . . 3 ⊢ (◡{〈2, 6〉} ∪ ◡{〈3, 9〉}) = ({〈6, 2〉} ∪ {〈9, 3〉}) |
13 | 1, 12 | eqtri 2754 | . 2 ⊢ ◡({〈2, 6〉} ∪ {〈3, 9〉}) = ({〈6, 2〉} ∪ {〈9, 3〉}) |
14 | df-pr 4626 | . . 3 ⊢ {〈2, 6〉, 〈3, 9〉} = ({〈2, 6〉} ∪ {〈3, 9〉}) | |
15 | 14 | cnveqi 5873 | . 2 ⊢ ◡{〈2, 6〉, 〈3, 9〉} = ◡({〈2, 6〉} ∪ {〈3, 9〉}) |
16 | df-pr 4626 | . 2 ⊢ {〈6, 2〉, 〈9, 3〉} = ({〈6, 2〉} ∪ {〈9, 3〉}) | |
17 | 13, 15, 16 | 3eqtr4i 2764 | 1 ⊢ ◡{〈2, 6〉, 〈3, 9〉} = {〈6, 2〉, 〈9, 3〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∪ cun 3944 {csn 4623 {cpr 4625 〈cop 4629 ◡ccnv 5673 ℕcn 12258 2c2 12313 3c3 12314 6c6 12317 9c9 12320 |
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 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 ax-1cn 11207 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 |
This theorem is referenced by: (None) |
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