Theorem resin 6877

Description: The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)

Assertion

Ref	Expression
resin	⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴⟶onto𝐶 ∧ (𝐹 ↾ 𝐵):𝐵⟶onto𝐷) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶1-1onto(𝐶 ∩ 𝐷))

Proof of Theorem resin

Step	Hyp	Ref	Expression
1		resdif 6876	. . . 4 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴⟶onto𝐶 ∧ (𝐹 ↾ 𝐵):𝐵⟶onto𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)⟶1-1onto(𝐶 ∖ 𝐷))
2		f1ofo 6862	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)⟶1-1onto(𝐶 ∖ 𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)⟶onto(𝐶 ∖ 𝐷))
3	1, 2	syl 18	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴⟶onto𝐶 ∧ (𝐹 ↾ 𝐵):𝐵⟶onto𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)⟶onto(𝐶 ∖ 𝐷))
4		resdif 6876	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴⟶onto𝐶 ∧ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)⟶onto(𝐶 ∖ 𝐷)) → (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)))
5	3, 4	syld3an3 1437	. 2 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴⟶onto𝐶 ∧ (𝐹 ↾ 𝐵):𝐵⟶onto𝐷) → (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)))
6		dfin4 4226	. . . 4 ⊢ (𝐶 ∩ 𝐷) = (𝐶 ∖ (𝐶 ∖ 𝐷))
7		f1oeq3 6844	. . . 4 ⊢ ((𝐶 ∩ 𝐷) = (𝐶 ∖ (𝐶 ∖ 𝐷)) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶1-1onto(𝐶 ∩ 𝐷) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷))))
8	6, 7	ax-mp 5	. . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶1-1onto(𝐶 ∩ 𝐷) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)))
9		dfin4 4226	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
10		f1oeq2 6843	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷))))
11	9, 10	ax-mp 5	. . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)))
12	9	reseq2i 5983	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵)))
13		f1oeq1 6842	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷))))
14	12, 13	ax-mp 5	. . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)))
15	8, 11, 14	3bitrri 301	. 2 ⊢ ((𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))⟶1-1onto(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶1-1onto(𝐶 ∩ 𝐷))
16	5, 15	sylib 221	1 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴⟶onto𝐶 ∧ (𝐹 ↾ 𝐵):𝐵⟶onto𝐷) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶1-1onto(𝐶 ∩ 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∖ cdif 4048   ∩ cin 4092  ^◡ccnv 5657   ↾ cres 5660  Fun wfun 6560  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573

This theorem is used by: (None)