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Mirrors > Home > MPE Home > Th. List > ex-cnv | Structured version Visualization version GIF version |
Description: Example for df-cnv 5686. 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 6147 | . . 3 ⊢ ◡({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = (◡{⟨2, 6⟩} ∪ ◡{⟨3, 9⟩}) | |
2 | 2nn 12316 | . . . . . 6 ⊢ 2 ∈ ℕ | |
3 | 2 | elexi 3491 | . . . . 5 ⊢ 2 ∈ V |
4 | 6nn 12332 | . . . . . 6 ⊢ 6 ∈ ℕ | |
5 | 4 | elexi 3491 | . . . . 5 ⊢ 6 ∈ V |
6 | 3, 5 | cnvsn 6230 | . . . 4 ⊢ ◡{⟨2, 6⟩} = {⟨6, 2⟩} |
7 | 3nn 12322 | . . . . . 6 ⊢ 3 ∈ ℕ | |
8 | 7 | elexi 3491 | . . . . 5 ⊢ 3 ∈ V |
9 | 9nn 12341 | . . . . . 6 ⊢ 9 ∈ ℕ | |
10 | 9 | elexi 3491 | . . . . 5 ⊢ 9 ∈ V |
11 | 8, 10 | cnvsn 6230 | . . . 4 ⊢ ◡{⟨3, 9⟩} = {⟨9, 3⟩} |
12 | 6, 11 | uneq12i 4160 | . . 3 ⊢ (◡{⟨2, 6⟩} ∪ ◡{⟨3, 9⟩}) = ({⟨6, 2⟩} ∪ {⟨9, 3⟩}) |
13 | 1, 12 | eqtri 2756 | . 2 ⊢ ◡({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = ({⟨6, 2⟩} ∪ {⟨9, 3⟩}) |
14 | df-pr 4632 | . . 3 ⊢ {⟨2, 6⟩, ⟨3, 9⟩} = ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) | |
15 | 14 | cnveqi 5877 | . 2 ⊢ ◡{⟨2, 6⟩, ⟨3, 9⟩} = ◡({⟨2, 6⟩} ∪ {⟨3, 9⟩}) |
16 | df-pr 4632 | . 2 ⊢ {⟨6, 2⟩, ⟨9, 3⟩} = ({⟨6, 2⟩} ∪ {⟨9, 3⟩}) | |
17 | 13, 15, 16 | 3eqtr4i 2766 | 1 ⊢ ◡{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∪ cun 3945 {csn 4629 {cpr 4631 ⟨cop 4635 ◡ccnv 5677 ℕcn 12243 2c2 12298 3c3 12299 6c6 12302 9c9 12305 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 ax-1cn 11197 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 |
This theorem is referenced by: (None) |
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