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Mirrors > Home > NFE Home > Th. List > 2elresin | GIF version |
Description: Membership in two functions restricted by each other's domain. (Contributed by set.mm contributors, 8-Aug-1994.) |
Ref | Expression |
---|---|
2elresin | ⊢ ((F Fn A ∧ G Fn B) → ((〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G) ↔ (〈x, y〉 ∈ (F ↾ (A ∩ B)) ∧ 〈x, z〉 ∈ (G ↾ (A ∩ B))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnop 5186 | . . . . . . 7 ⊢ ((F Fn A ∧ 〈x, y〉 ∈ F) → x ∈ A) | |
2 | fnop 5186 | . . . . . . 7 ⊢ ((G Fn B ∧ 〈x, z〉 ∈ G) → x ∈ B) | |
3 | 1, 2 | anim12i 549 | . . . . . 6 ⊢ (((F Fn A ∧ 〈x, y〉 ∈ F) ∧ (G Fn B ∧ 〈x, z〉 ∈ G)) → (x ∈ A ∧ x ∈ B)) |
4 | an4 797 | . . . . . 6 ⊢ (((F Fn A ∧ G Fn B) ∧ (〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G)) ↔ ((F Fn A ∧ 〈x, y〉 ∈ F) ∧ (G Fn B ∧ 〈x, z〉 ∈ G))) | |
5 | elin 3219 | . . . . . 6 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B)) | |
6 | 3, 4, 5 | 3imtr4i 257 | . . . . 5 ⊢ (((F Fn A ∧ G Fn B) ∧ (〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G)) → x ∈ (A ∩ B)) |
7 | opelres 4950 | . . . . . . 7 ⊢ (〈x, y〉 ∈ (F ↾ (A ∩ B)) ↔ (〈x, y〉 ∈ F ∧ x ∈ (A ∩ B))) | |
8 | 7 | simplbi2com 1374 | . . . . . 6 ⊢ (x ∈ (A ∩ B) → (〈x, y〉 ∈ F → 〈x, y〉 ∈ (F ↾ (A ∩ B)))) |
9 | opelres 4950 | . . . . . . 7 ⊢ (〈x, z〉 ∈ (G ↾ (A ∩ B)) ↔ (〈x, z〉 ∈ G ∧ x ∈ (A ∩ B))) | |
10 | 9 | simplbi2com 1374 | . . . . . 6 ⊢ (x ∈ (A ∩ B) → (〈x, z〉 ∈ G → 〈x, z〉 ∈ (G ↾ (A ∩ B)))) |
11 | 8, 10 | anim12d 546 | . . . . 5 ⊢ (x ∈ (A ∩ B) → ((〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G) → (〈x, y〉 ∈ (F ↾ (A ∩ B)) ∧ 〈x, z〉 ∈ (G ↾ (A ∩ B))))) |
12 | 6, 11 | syl 15 | . . . 4 ⊢ (((F Fn A ∧ G Fn B) ∧ (〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G)) → ((〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G) → (〈x, y〉 ∈ (F ↾ (A ∩ B)) ∧ 〈x, z〉 ∈ (G ↾ (A ∩ B))))) |
13 | 12 | ex 423 | . . 3 ⊢ ((F Fn A ∧ G Fn B) → ((〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G) → ((〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G) → (〈x, y〉 ∈ (F ↾ (A ∩ B)) ∧ 〈x, z〉 ∈ (G ↾ (A ∩ B)))))) |
14 | 13 | pm2.43d 44 | . 2 ⊢ ((F Fn A ∧ G Fn B) → ((〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G) → (〈x, y〉 ∈ (F ↾ (A ∩ B)) ∧ 〈x, z〉 ∈ (G ↾ (A ∩ B))))) |
15 | resss 4988 | . . . 4 ⊢ (F ↾ (A ∩ B)) ⊆ F | |
16 | 15 | sseli 3269 | . . 3 ⊢ (〈x, y〉 ∈ (F ↾ (A ∩ B)) → 〈x, y〉 ∈ F) |
17 | resss 4988 | . . . 4 ⊢ (G ↾ (A ∩ B)) ⊆ G | |
18 | 17 | sseli 3269 | . . 3 ⊢ (〈x, z〉 ∈ (G ↾ (A ∩ B)) → 〈x, z〉 ∈ G) |
19 | 16, 18 | anim12i 549 | . 2 ⊢ ((〈x, y〉 ∈ (F ↾ (A ∩ B)) ∧ 〈x, z〉 ∈ (G ↾ (A ∩ B))) → (〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G)) |
20 | 14, 19 | impbid1 194 | 1 ⊢ ((F Fn A ∧ G Fn B) → ((〈x, y〉 ∈ F ∧ 〈x, z〉 ∈ G) ↔ (〈x, y〉 ∈ (F ↾ (A ∩ B)) ∧ 〈x, z〉 ∈ (G ↾ (A ∩ B))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∈ wcel 1710 ∩ cin 3208 〈cop 4561 ↾ cres 4774 Fn wfn 4776 |
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-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fn 4790 |
This theorem is referenced by: (None) |
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