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Mirrors > Home > NFE Home > Th. List > br2nd | GIF version |
Description: Binary relationship equivalence for the 2nd function. (Contributed by set.mm contributors, 8-Jan-2015.) |
Ref | Expression |
---|---|
br1st.1 | ⊢ B ∈ V |
Ref | Expression |
---|---|
br2nd | ⊢ (A2nd B ↔ ∃x A = 〈x, B〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brex 4689 | . . 3 ⊢ (A2nd B → (A ∈ V ∧ B ∈ V)) | |
2 | 1 | simpld 445 | . 2 ⊢ (A2nd B → A ∈ V) |
3 | vex 2862 | . . . . 5 ⊢ x ∈ V | |
4 | br1st.1 | . . . . 5 ⊢ B ∈ V | |
5 | 3, 4 | opex 4588 | . . . 4 ⊢ 〈x, B〉 ∈ V |
6 | eleq1 2413 | . . . 4 ⊢ (A = 〈x, B〉 → (A ∈ V ↔ 〈x, B〉 ∈ V)) | |
7 | 5, 6 | mpbiri 224 | . . 3 ⊢ (A = 〈x, B〉 → A ∈ V) |
8 | 7 | exlimiv 1634 | . 2 ⊢ (∃x A = 〈x, B〉 → A ∈ V) |
9 | eqeq1 2359 | . . . . 5 ⊢ (y = A → (y = 〈x, z〉 ↔ A = 〈x, z〉)) | |
10 | 9 | exbidv 1626 | . . . 4 ⊢ (y = A → (∃x y = 〈x, z〉 ↔ ∃x A = 〈x, z〉)) |
11 | opeq2 4579 | . . . . . 6 ⊢ (z = B → 〈x, z〉 = 〈x, B〉) | |
12 | 11 | eqeq2d 2364 | . . . . 5 ⊢ (z = B → (A = 〈x, z〉 ↔ A = 〈x, B〉)) |
13 | 12 | exbidv 1626 | . . . 4 ⊢ (z = B → (∃x A = 〈x, z〉 ↔ ∃x A = 〈x, B〉)) |
14 | df-2nd 4797 | . . . 4 ⊢ 2nd = {〈y, z〉 ∣ ∃x y = 〈x, z〉} | |
15 | 10, 13, 14 | brabg 4706 | . . 3 ⊢ ((A ∈ V ∧ B ∈ V) → (A2nd B ↔ ∃x A = 〈x, B〉)) |
16 | 4, 15 | mpan2 652 | . 2 ⊢ (A ∈ V → (A2nd B ↔ ∃x A = 〈x, B〉)) |
17 | 2, 8, 16 | pm5.21nii 342 | 1 ⊢ (A2nd B ↔ ∃x A = 〈x, B〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 〈cop 4561 class class class wbr 4639 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-2nd 4797 |
This theorem is referenced by: dfxp2 5113 opbr2nd 5502 2ndfo 5506 dfrn5 5508 brtxp 5783 |
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