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Mirrors > Home > MPE Home > Th. List > ex-cnv | Structured version Visualization version GIF version |
Description: Example for df-cnv 5563. 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 6001 | . . 3 ⊢ ◡({〈2, 6〉} ∪ {〈3, 9〉}) = (◡{〈2, 6〉} ∪ ◡{〈3, 9〉}) | |
2 | 2nn 11711 | . . . . . 6 ⊢ 2 ∈ ℕ | |
3 | 2 | elexi 3513 | . . . . 5 ⊢ 2 ∈ V |
4 | 6nn 11727 | . . . . . 6 ⊢ 6 ∈ ℕ | |
5 | 4 | elexi 3513 | . . . . 5 ⊢ 6 ∈ V |
6 | 3, 5 | cnvsn 6083 | . . . 4 ⊢ ◡{〈2, 6〉} = {〈6, 2〉} |
7 | 3nn 11717 | . . . . . 6 ⊢ 3 ∈ ℕ | |
8 | 7 | elexi 3513 | . . . . 5 ⊢ 3 ∈ V |
9 | 9nn 11736 | . . . . . 6 ⊢ 9 ∈ ℕ | |
10 | 9 | elexi 3513 | . . . . 5 ⊢ 9 ∈ V |
11 | 8, 10 | cnvsn 6083 | . . . 4 ⊢ ◡{〈3, 9〉} = {〈9, 3〉} |
12 | 6, 11 | uneq12i 4137 | . . 3 ⊢ (◡{〈2, 6〉} ∪ ◡{〈3, 9〉}) = ({〈6, 2〉} ∪ {〈9, 3〉}) |
13 | 1, 12 | eqtri 2844 | . 2 ⊢ ◡({〈2, 6〉} ∪ {〈3, 9〉}) = ({〈6, 2〉} ∪ {〈9, 3〉}) |
14 | df-pr 4570 | . . 3 ⊢ {〈2, 6〉, 〈3, 9〉} = ({〈2, 6〉} ∪ {〈3, 9〉}) | |
15 | 14 | cnveqi 5745 | . 2 ⊢ ◡{〈2, 6〉, 〈3, 9〉} = ◡({〈2, 6〉} ∪ {〈3, 9〉}) |
16 | df-pr 4570 | . 2 ⊢ {〈6, 2〉, 〈9, 3〉} = ({〈6, 2〉} ∪ {〈9, 3〉}) | |
17 | 13, 15, 16 | 3eqtr4i 2854 | 1 ⊢ ◡{〈2, 6〉, 〈3, 9〉} = {〈6, 2〉, 〈9, 3〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3934 {csn 4567 {cpr 4569 〈cop 4573 ◡ccnv 5554 ℕcn 11638 2c2 11693 3c3 11694 6c6 11697 9c9 11700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-1cn 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 |
This theorem is referenced by: (None) |
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