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Mirrors > Home > NFE Home > Th. List > df2nd2 | GIF version |
Description: Alternate definition of the 2nd function. (Contributed by SF, 8-Jan-2015.) |
Ref | Expression |
---|---|
df2nd2 | ⊢ 2nd = (1st ∘ Swap ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 | . . . . . . . 8 ⊢ y ∈ V | |
2 | 1 | br1st 4858 | . . . . . . 7 ⊢ (w1st y ↔ ∃z w = 〈y, z〉) |
3 | 2 | anbi1i 676 | . . . . . 6 ⊢ ((w1st y ∧ x Swap w) ↔ (∃z w = 〈y, z〉 ∧ x Swap w)) |
4 | ancom 437 | . . . . . 6 ⊢ ((x Swap w ∧ w1st y) ↔ (w1st y ∧ x Swap w)) | |
5 | 19.41v 1901 | . . . . . 6 ⊢ (∃z(w = 〈y, z〉 ∧ x Swap w) ↔ (∃z w = 〈y, z〉 ∧ x Swap w)) | |
6 | 3, 4, 5 | 3bitr4i 268 | . . . . 5 ⊢ ((x Swap w ∧ w1st y) ↔ ∃z(w = 〈y, z〉 ∧ x Swap w)) |
7 | 6 | exbii 1582 | . . . 4 ⊢ (∃w(x Swap w ∧ w1st y) ↔ ∃w∃z(w = 〈y, z〉 ∧ x Swap w)) |
8 | excom 1741 | . . . 4 ⊢ (∃z∃w(w = 〈y, z〉 ∧ x Swap w) ↔ ∃w∃z(w = 〈y, z〉 ∧ x Swap w)) | |
9 | vex 2862 | . . . . . . . 8 ⊢ z ∈ V | |
10 | 1, 9 | opex 4588 | . . . . . . 7 ⊢ 〈y, z〉 ∈ V |
11 | breq2 4643 | . . . . . . 7 ⊢ (w = 〈y, z〉 → (x Swap w ↔ x Swap 〈y, z〉)) | |
12 | 10, 11 | ceqsexv 2894 | . . . . . 6 ⊢ (∃w(w = 〈y, z〉 ∧ x Swap w) ↔ x Swap 〈y, z〉) |
13 | 1, 9 | brswap2 4860 | . . . . . 6 ⊢ (x Swap 〈y, z〉 ↔ x = 〈z, y〉) |
14 | 12, 13 | bitri 240 | . . . . 5 ⊢ (∃w(w = 〈y, z〉 ∧ x Swap w) ↔ x = 〈z, y〉) |
15 | 14 | exbii 1582 | . . . 4 ⊢ (∃z∃w(w = 〈y, z〉 ∧ x Swap w) ↔ ∃z x = 〈z, y〉) |
16 | 7, 8, 15 | 3bitr2ri 265 | . . 3 ⊢ (∃z x = 〈z, y〉 ↔ ∃w(x Swap w ∧ w1st y)) |
17 | 16 | opabbii 4626 | . 2 ⊢ {〈x, y〉 ∣ ∃z x = 〈z, y〉} = {〈x, y〉 ∣ ∃w(x Swap w ∧ w1st y)} |
18 | df-2nd 4797 | . 2 ⊢ 2nd = {〈x, y〉 ∣ ∃z x = 〈z, y〉} | |
19 | df-co 4726 | . 2 ⊢ (1st ∘ Swap ) = {〈x, y〉 ∣ ∃w(x Swap w ∧ w1st y)} | |
20 | 17, 18, 19 | 3eqtr4i 2383 | 1 ⊢ 2nd = (1st ∘ Swap ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 〈cop 4561 {copab 4622 class class class wbr 4639 1st c1st 4717 Swap cswap 4718 ∘ ccom 4721 2nd c2nd 4783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-co 4726 df-2nd 4797 |
This theorem is referenced by: 2ndex 5112 |
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