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| Description: Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| 2nd1st | ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 1st2nd2 8053 | . . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
| 2 | 1 | sneqd 4638 | . . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {〈(1st ‘𝐴), (2nd ‘𝐴)〉}) | 
| 3 | 2 | cnveqd 5886 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) | 
| 4 | 3 | unieqd 4920 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) | 
| 5 | opswap 6249 | . 2 ⊢ ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉 | |
| 6 | 4, 5 | eqtrdi 2793 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 ∪ cuni 4907 × cxp 5683 ◡ccnv 5684 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 | 
| This theorem is referenced by: fcnvgreu 32683 gsumhashmul 33064 tposideq 48788 swapf1a 48975 swapf2a 48977 | 
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