Theorem resin 6840

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 6839	. . . 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 6825	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐶 ∖ 𝐷))
3	1, 2	syl 18	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐶 ∖ 𝐷))
4		resdif 6839	. . 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 1436	. 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 4224	. . . 4 ⊢ (𝐶 ∩ 𝐷) = (𝐶 ∖ (𝐶 ∖ 𝐷))
7		f1oeq3 6807	. . . 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 4224	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
10		f1oeq2 6806	. . . 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 5969	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵)))
13		f1oeq1 6805	. . . 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 301	. 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 221	1 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∖ cdif 3896   ∩ cin 3898  ^◡ccnv 5654   ↾ cres 5657  Fun wfun 6527  –onto→wfo 6531  –1-1-onto→wf1o 6532

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732  ax-sep 5251  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 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540

This theorem is used by: (None)