Theorem 2elresin 5194

Description: Membership in two functions restricted by each other's domain. (Contributed by set.mm contributors, 8-Aug-1994.)

Assertion

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)))))

Proof of Theorem 2elresin

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)))))

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)