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Mirrors > Home > NFE Home > Th. List > ssrel | GIF version |
Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by NM, 2-Aug-1994.) (Revised by set.mm contributors, 27-Aug-2011.) |
Ref | Expression |
---|---|
ssrel | ⊢ (A ⊆ B ↔ ∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3267 | . . 3 ⊢ (A ⊆ B → (〈x, y〉 ∈ A → 〈x, y〉 ∈ B)) | |
2 | 1 | alrimivv 1632 | . 2 ⊢ (A ⊆ B → ∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B)) |
3 | vex 2862 | . . . . . . 7 ⊢ z ∈ V | |
4 | 3 | proj1ex 4593 | . . . . . 6 ⊢ Proj1 z ∈ V |
5 | opeq1 4578 | . . . . . . . . 9 ⊢ (x = Proj1 z → 〈x, y〉 = 〈 Proj1 z, y〉) | |
6 | 5 | eleq1d 2419 | . . . . . . . 8 ⊢ (x = Proj1 z → (〈x, y〉 ∈ A ↔ 〈 Proj1 z, y〉 ∈ A)) |
7 | 5 | eleq1d 2419 | . . . . . . . 8 ⊢ (x = Proj1 z → (〈x, y〉 ∈ B ↔ 〈 Proj1 z, y〉 ∈ B)) |
8 | 6, 7 | imbi12d 311 | . . . . . . 7 ⊢ (x = Proj1 z → ((〈x, y〉 ∈ A → 〈x, y〉 ∈ B) ↔ (〈 Proj1 z, y〉 ∈ A → 〈 Proj1 z, y〉 ∈ B))) |
9 | 8 | albidv 1625 | . . . . . 6 ⊢ (x = Proj1 z → (∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B) ↔ ∀y(〈 Proj1 z, y〉 ∈ A → 〈 Proj1 z, y〉 ∈ B))) |
10 | 4, 9 | spcv 2945 | . . . . 5 ⊢ (∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B) → ∀y(〈 Proj1 z, y〉 ∈ A → 〈 Proj1 z, y〉 ∈ B)) |
11 | 3 | proj2ex 4594 | . . . . . 6 ⊢ Proj2 z ∈ V |
12 | opeq2 4579 | . . . . . . . 8 ⊢ (y = Proj2 z → 〈 Proj1 z, y〉 = 〈 Proj1 z, Proj2 z〉) | |
13 | 12 | eleq1d 2419 | . . . . . . 7 ⊢ (y = Proj2 z → (〈 Proj1 z, y〉 ∈ A ↔ 〈 Proj1 z, Proj2 z〉 ∈ A)) |
14 | 12 | eleq1d 2419 | . . . . . . 7 ⊢ (y = Proj2 z → (〈 Proj1 z, y〉 ∈ B ↔ 〈 Proj1 z, Proj2 z〉 ∈ B)) |
15 | 13, 14 | imbi12d 311 | . . . . . 6 ⊢ (y = Proj2 z → ((〈 Proj1 z, y〉 ∈ A → 〈 Proj1 z, y〉 ∈ B) ↔ (〈 Proj1 z, Proj2 z〉 ∈ A → 〈 Proj1 z, Proj2 z〉 ∈ B))) |
16 | 11, 15 | spcv 2945 | . . . . 5 ⊢ (∀y(〈 Proj1 z, y〉 ∈ A → 〈 Proj1 z, y〉 ∈ B) → (〈 Proj1 z, Proj2 z〉 ∈ A → 〈 Proj1 z, Proj2 z〉 ∈ B)) |
17 | 10, 16 | syl 15 | . . . 4 ⊢ (∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B) → (〈 Proj1 z, Proj2 z〉 ∈ A → 〈 Proj1 z, Proj2 z〉 ∈ B)) |
18 | opeq 4619 | . . . . 5 ⊢ z = 〈 Proj1 z, Proj2 z〉 | |
19 | 18 | eleq1i 2416 | . . . 4 ⊢ (z ∈ A ↔ 〈 Proj1 z, Proj2 z〉 ∈ A) |
20 | 18 | eleq1i 2416 | . . . 4 ⊢ (z ∈ B ↔ 〈 Proj1 z, Proj2 z〉 ∈ B) |
21 | 17, 19, 20 | 3imtr4g 261 | . . 3 ⊢ (∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B) → (z ∈ A → z ∈ B)) |
22 | 21 | ssrdv 3278 | . 2 ⊢ (∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B) → A ⊆ B) |
23 | 2, 22 | impbii 180 | 1 ⊢ (A ⊆ B ↔ ∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 = wceq 1642 ∈ wcel 1710 ⊆ wss 3257 〈cop 4561 Proj1 cproj1 4563 Proj2 cproj2 4564 |
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-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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-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-0c 4377 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 |
This theorem is referenced by: eqrel 4845 ssopr 4846 relssi 4848 relssdv 4849 cotr 5026 cnvsym 5027 intasym 5028 intirr 5029 ssdmrn 5099 dffun2 5119 fvfullfunlem2 5862 |
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