Theorem resin 6796

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 6795	. . . 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 6781	. . . 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 6795	. . 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 4230	. . . 4 ⊢ (𝐶 ∩ 𝐷) = (𝐶 ∖ (𝐶 ∖ 𝐷))
7		f1oeq3 6764	. . . 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 4230	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
10		f1oeq2 6763	. . . 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 5935	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵)))
13		f1oeq1 6762	. . . 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 1541   ∖ cdif 3898   ∩ cin 3900  ^◡ccnv 5623   ↾ cres 5626  Fun wfun 6486  –onto→wfo 6490  –1-1-onto→wf1o 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499

This theorem is referenced by: (None)