Theorem resin 6845

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–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷))

Proof of Theorem resin

Step	Hyp	Ref	Expression
1		resdif 6844	. . . 4 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷))
2		f1ofo 6830	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐶 ∖ 𝐷))
3	1, 2	syl 17	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐶 ∖ 𝐷))
4		resdif 6844	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐶 ∖ 𝐷)) → (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)))
5	3, 4	syld3an3 1411	. 2 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)))
6		dfin4 4258	. . . 4 ⊢ (𝐶 ∩ 𝐷) = (𝐶 ∖ (𝐶 ∖ 𝐷))
7		f1oeq3 6813	. . . 4 ⊢ ((𝐶 ∩ 𝐷) = (𝐶 ∖ (𝐶 ∖ 𝐷)) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷))))
8	6, 7	ax-mp 5	. . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)))
9		dfin4 4258	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
10		f1oeq2 6812	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷))))
11	9, 10	ax-mp 5	. . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)))
12	9	reseq2i 5968	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵)))
13		f1oeq1 6811	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷))))
14	12, 13	ax-mp 5	. . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)))
15	8, 11, 14	3bitrri 298	. 2 ⊢ ((𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷))
16	5, 15	sylib 218	1 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∖ cdif 3928   ∩ cin 3930  ^◡ccnv 5658   ↾ cres 5661  Fun wfun 6530  –onto→wfo 6534  –1-1-onto→wf1o 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543

This theorem is referenced by: (None)