Theorem resin 6721

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 6720	. . . 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 6707	. . . 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 6720	. . 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 1407	. 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 4198	. . . 4 ⊢ (𝐶 ∩ 𝐷) = (𝐶 ∖ (𝐶 ∖ 𝐷))
7		f1oeq3 6690	. . . 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 4198	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
10		f1oeq2 6689	. . . 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 5877	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵)))
13		f1oeq1 6688	. . . 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 297	. 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 217	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 205   ∧ w3a 1085   = wceq 1539   ∖ cdif 3880   ∩ cin 3882  ^◡ccnv 5579   ↾ cres 5582  Fun wfun 6412  –onto→wfo 6416  –1-1-onto→wf1o 6417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425

This theorem is referenced by: (None)