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Mirrors > Home > NFE Home > Th. List > enprmaplem4 | GIF version |
Description: Lemma for enprmap 6082. More stratification condition setup. (Contributed by SF, 3-Mar-2015.) |
Ref | Expression |
---|---|
enprmaplem4.1 | ⊢ R = (u ∈ B ↦ if(u ∈ p, x, y)) |
enprmaplem4.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
enprmaplem4 | ⊢ R ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enprmaplem4.1 | . . 3 ⊢ R = (u ∈ B ↦ if(u ∈ p, x, y)) | |
2 | elun 3220 | . . . . . 6 ⊢ (〈u, {z}〉 ∈ ((p × ℘1x) ∪ ( ∼ p × ℘1y)) ↔ (〈u, {z}〉 ∈ (p × ℘1x) ∨ 〈u, {z}〉 ∈ ( ∼ p × ℘1y))) | |
3 | opelxp 4811 | . . . . . . . 8 ⊢ (〈u, {z}〉 ∈ (p × ℘1x) ↔ (u ∈ p ∧ {z} ∈ ℘1x)) | |
4 | snelpw1 4146 | . . . . . . . . 9 ⊢ ({z} ∈ ℘1x ↔ z ∈ x) | |
5 | 4 | anbi2i 675 | . . . . . . . 8 ⊢ ((u ∈ p ∧ {z} ∈ ℘1x) ↔ (u ∈ p ∧ z ∈ x)) |
6 | 3, 5 | bitri 240 | . . . . . . 7 ⊢ (〈u, {z}〉 ∈ (p × ℘1x) ↔ (u ∈ p ∧ z ∈ x)) |
7 | opelxp 4811 | . . . . . . . 8 ⊢ (〈u, {z}〉 ∈ ( ∼ p × ℘1y) ↔ (u ∈ ∼ p ∧ {z} ∈ ℘1y)) | |
8 | vex 2862 | . . . . . . . . . 10 ⊢ u ∈ V | |
9 | 8 | elcompl 3225 | . . . . . . . . 9 ⊢ (u ∈ ∼ p ↔ ¬ u ∈ p) |
10 | snelpw1 4146 | . . . . . . . . 9 ⊢ ({z} ∈ ℘1y ↔ z ∈ y) | |
11 | 9, 10 | anbi12i 678 | . . . . . . . 8 ⊢ ((u ∈ ∼ p ∧ {z} ∈ ℘1y) ↔ (¬ u ∈ p ∧ z ∈ y)) |
12 | 7, 11 | bitri 240 | . . . . . . 7 ⊢ (〈u, {z}〉 ∈ ( ∼ p × ℘1y) ↔ (¬ u ∈ p ∧ z ∈ y)) |
13 | 6, 12 | orbi12i 507 | . . . . . 6 ⊢ ((〈u, {z}〉 ∈ (p × ℘1x) ∨ 〈u, {z}〉 ∈ ( ∼ p × ℘1y)) ↔ ((u ∈ p ∧ z ∈ x) ∨ (¬ u ∈ p ∧ z ∈ y))) |
14 | 2, 13 | bitri 240 | . . . . 5 ⊢ (〈u, {z}〉 ∈ ((p × ℘1x) ∪ ( ∼ p × ℘1y)) ↔ ((u ∈ p ∧ z ∈ x) ∨ (¬ u ∈ p ∧ z ∈ y))) |
15 | opelcnv 4893 | . . . . 5 ⊢ (〈{z}, u〉 ∈ ◡((p × ℘1x) ∪ ( ∼ p × ℘1y)) ↔ 〈u, {z}〉 ∈ ((p × ℘1x) ∪ ( ∼ p × ℘1y))) | |
16 | elif 3696 | . . . . 5 ⊢ (z ∈ if(u ∈ p, x, y) ↔ ((u ∈ p ∧ z ∈ x) ∨ (¬ u ∈ p ∧ z ∈ y))) | |
17 | 14, 15, 16 | 3bitr4i 268 | . . . 4 ⊢ (〈{z}, u〉 ∈ ◡((p × ℘1x) ∪ ( ∼ p × ℘1y)) ↔ z ∈ if(u ∈ p, x, y)) |
18 | 17 | releqmpt 5808 | . . 3 ⊢ ((B × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ◡((p × ℘1x) ∪ ( ∼ p × ℘1y))) “ 1c)) = (u ∈ B ↦ if(u ∈ p, x, y)) |
19 | 1, 18 | eqtr4i 2376 | . 2 ⊢ R = ((B × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ◡((p × ℘1x) ∪ ( ∼ p × ℘1y))) “ 1c)) |
20 | enprmaplem4.2 | . . 3 ⊢ B ∈ V | |
21 | vex 2862 | . . . . . 6 ⊢ p ∈ V | |
22 | vex 2862 | . . . . . . 7 ⊢ x ∈ V | |
23 | 22 | pw1ex 4303 | . . . . . 6 ⊢ ℘1x ∈ V |
24 | 21, 23 | xpex 5115 | . . . . 5 ⊢ (p × ℘1x) ∈ V |
25 | 21 | complex 4104 | . . . . . 6 ⊢ ∼ p ∈ V |
26 | vex 2862 | . . . . . . 7 ⊢ y ∈ V | |
27 | 26 | pw1ex 4303 | . . . . . 6 ⊢ ℘1y ∈ V |
28 | 25, 27 | xpex 5115 | . . . . 5 ⊢ ( ∼ p × ℘1y) ∈ V |
29 | 24, 28 | unex 4106 | . . . 4 ⊢ ((p × ℘1x) ∪ ( ∼ p × ℘1y)) ∈ V |
30 | 29 | cnvex 5102 | . . 3 ⊢ ◡((p × ℘1x) ∪ ( ∼ p × ℘1y)) ∈ V |
31 | 20, 30 | mptexlem 5810 | . 2 ⊢ ((B × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ◡((p × ℘1x) ∪ ( ∼ p × ℘1y))) “ 1c)) ∈ V |
32 | 19, 31 | eqeltri 2423 | 1 ⊢ R ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 357 ∧ wa 358 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 ∪ cun 3207 ∩ cin 3208 ⊕ csymdif 3209 ifcif 3662 {csn 3737 1cc1c 4134 ℘1cpw1 4135 〈cop 4561 S csset 4719 “ cima 4722 × cxp 4770 ◡ccnv 4771 ↦ cmpt 5651 Ins2 cins2 5749 Ins3 cins3 5751 |
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-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-xp 4784 df-cnv 4785 df-2nd 4797 df-mpt 5652 df-txp 5736 df-ins2 5750 df-ins3 5752 |
This theorem is referenced by: enprmaplem5 6080 |
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