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Mirrors > Home > NFE Home > Th. List > oprssdm | GIF version |
Description: Domain of closure of an operation. (Contributed by set.mm contributors, 24-Aug-1995.) |
Ref | Expression |
---|---|
oprssdm.1 | ⊢ ¬ ∅ ∈ S |
oprssdm.2 | ⊢ ((x ∈ S ∧ y ∈ S) → (xFy) ∈ S) |
Ref | Expression |
---|---|
oprssdm | ⊢ (S × S) ⊆ dom F |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 4811 | . . 3 ⊢ (〈x, y〉 ∈ (S × S) ↔ (x ∈ S ∧ y ∈ S)) | |
2 | ndmfv 5349 | . . . . 5 ⊢ (¬ 〈x, y〉 ∈ dom F → (F ‘〈x, y〉) = ∅) | |
3 | df-ov 5526 | . . . . . . . 8 ⊢ (xFy) = (F ‘〈x, y〉) | |
4 | 3 | eqeq1i 2360 | . . . . . . 7 ⊢ ((xFy) = ∅ ↔ (F ‘〈x, y〉) = ∅) |
5 | oprssdm.1 | . . . . . . . 8 ⊢ ¬ ∅ ∈ S | |
6 | eleq1 2413 | . . . . . . . 8 ⊢ ((xFy) = ∅ → ((xFy) ∈ S ↔ ∅ ∈ S)) | |
7 | 5, 6 | mtbiri 294 | . . . . . . 7 ⊢ ((xFy) = ∅ → ¬ (xFy) ∈ S) |
8 | 4, 7 | sylbir 204 | . . . . . 6 ⊢ ((F ‘〈x, y〉) = ∅ → ¬ (xFy) ∈ S) |
9 | oprssdm.2 | . . . . . 6 ⊢ ((x ∈ S ∧ y ∈ S) → (xFy) ∈ S) | |
10 | 8, 9 | nsyl 113 | . . . . 5 ⊢ ((F ‘〈x, y〉) = ∅ → ¬ (x ∈ S ∧ y ∈ S)) |
11 | 2, 10 | syl 15 | . . . 4 ⊢ (¬ 〈x, y〉 ∈ dom F → ¬ (x ∈ S ∧ y ∈ S)) |
12 | 11 | con4i 122 | . . 3 ⊢ ((x ∈ S ∧ y ∈ S) → 〈x, y〉 ∈ dom F) |
13 | 1, 12 | sylbi 187 | . 2 ⊢ (〈x, y〉 ∈ (S × S) → 〈x, y〉 ∈ dom F) |
14 | 13 | relssi 4848 | 1 ⊢ (S × S) ⊆ dom F |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ⊆ wss 3257 ∅c0 3550 〈cop 4561 × cxp 4770 dom cdm 4772 ‘cfv 4781 (class class class)co 5525 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-fv 4795 df-ov 5526 |
This theorem is referenced by: (None) |
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