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Mirrors > Home > MPE Home > Th. List > ex-cnv | Structured version Visualization version GIF version |
Description: Example for df-cnv 5677. 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 6135 | . . 3 ⊢ ◡({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = (◡{⟨2, 6⟩} ∪ ◡{⟨3, 9⟩}) | |
2 | 2nn 12286 | . . . . . 6 ⊢ 2 ∈ ℕ | |
3 | 2 | elexi 3488 | . . . . 5 ⊢ 2 ∈ V |
4 | 6nn 12302 | . . . . . 6 ⊢ 6 ∈ ℕ | |
5 | 4 | elexi 3488 | . . . . 5 ⊢ 6 ∈ V |
6 | 3, 5 | cnvsn 6218 | . . . 4 ⊢ ◡{⟨2, 6⟩} = {⟨6, 2⟩} |
7 | 3nn 12292 | . . . . . 6 ⊢ 3 ∈ ℕ | |
8 | 7 | elexi 3488 | . . . . 5 ⊢ 3 ∈ V |
9 | 9nn 12311 | . . . . . 6 ⊢ 9 ∈ ℕ | |
10 | 9 | elexi 3488 | . . . . 5 ⊢ 9 ∈ V |
11 | 8, 10 | cnvsn 6218 | . . . 4 ⊢ ◡{⟨3, 9⟩} = {⟨9, 3⟩} |
12 | 6, 11 | uneq12i 4156 | . . 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 5867 | . 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 1533 ∪ cun 3941 {csn 4623 {cpr 4625 ⟨cop 4629 ◡ccnv 5668 ℕcn 12213 2c2 12268 3c3 12269 6c6 12272 9c9 12275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 ax-1cn 11167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 |
This theorem is referenced by: (None) |
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