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Mirrors > Home > NFE Home > Th. List > resin | GIF version |
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.) |
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)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resdif 5306 | . . . 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 5293 | . . . 4 ⊢ ((F ↾ (A ∖ B)):(A ∖ B)–1-1-onto→(C ∖ D) → (F ↾ (A ∖ B)):(A ∖ B)–onto→(C ∖ D)) | |
3 | 1, 2 | syl 15 | . . 3 ⊢ ((Fun ◡F ∧ (F ↾ A):A–onto→C ∧ (F ↾ B):B–onto→D) → (F ↾ (A ∖ B)):(A ∖ B)–onto→(C ∖ D)) |
4 | resdif 5306 | . . 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 1227 | . 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 3495 | . . . 4 ⊢ (C ∩ D) = (C ∖ (C ∖ D)) | |
7 | f1oeq3 5283 | . . . 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 8 | . . 3 ⊢ ((F ↾ (A ∩ B)):(A ∩ B)–1-1-onto→(C ∩ D) ↔ (F ↾ (A ∩ B)):(A ∩ B)–1-1-onto→(C ∖ (C ∖ D))) |
9 | dfin4 3495 | . . . 4 ⊢ (A ∩ B) = (A ∖ (A ∖ B)) | |
10 | f1oeq2 5282 | . . . 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 8 | . . 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 4931 | . . . 4 ⊢ (F ↾ (A ∩ B)) = (F ↾ (A ∖ (A ∖ B))) |
13 | f1oeq1 5281 | . . . 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 8 | . . 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 263 | . 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 188 | 1 ⊢ ((Fun ◡F ∧ (F ↾ A):A–onto→C ∧ (F ↾ B):B–onto→D) → (F ↾ (A ∩ B)):(A ∩ B)–1-1-onto→(C ∩ D)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ w3a 934 = wceq 1642 ∖ cdif 3206 ∩ cin 3208 ◡ccnv 4771 ↾ cres 4774 Fun wfun 4775 –onto→wfo 4779 –1-1-onto→wf1o 4780 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 |
This theorem is referenced by: (None) |
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