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Mirrors > Home > NFE Home > Th. List > fores | GIF version |
Description: Restriction of a function. (Contributed by set.mm contributors, 4-Mar-1997.) |
Ref | Expression |
---|---|
fores | ⊢ ((Fun F ∧ A ⊆ dom F) → (F ↾ A):A–onto→(F “ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres 5143 | . . 3 ⊢ (Fun F → Fun (F ↾ A)) | |
2 | 1 | anim1i 551 | . 2 ⊢ ((Fun F ∧ A ⊆ dom F) → (Fun (F ↾ A) ∧ A ⊆ dom F)) |
3 | df-fn 4790 | . . 3 ⊢ ((F ↾ A) Fn A ↔ (Fun (F ↾ A) ∧ dom (F ↾ A) = A)) | |
4 | dfima3 4951 | . . . . 5 ⊢ (F “ A) = ran (F ↾ A) | |
5 | 4 | eqcomi 2357 | . . . 4 ⊢ ran (F ↾ A) = (F “ A) |
6 | df-fo 4793 | . . . 4 ⊢ ((F ↾ A):A–onto→(F “ A) ↔ ((F ↾ A) Fn A ∧ ran (F ↾ A) = (F “ A))) | |
7 | 5, 6 | mpbiran2 885 | . . 3 ⊢ ((F ↾ A):A–onto→(F “ A) ↔ (F ↾ A) Fn A) |
8 | ssdmres 4987 | . . . 4 ⊢ (A ⊆ dom F ↔ dom (F ↾ A) = A) | |
9 | 8 | anbi2i 675 | . . 3 ⊢ ((Fun (F ↾ A) ∧ A ⊆ dom F) ↔ (Fun (F ↾ A) ∧ dom (F ↾ A) = A)) |
10 | 3, 7, 9 | 3bitr4i 268 | . 2 ⊢ ((F ↾ A):A–onto→(F “ A) ↔ (Fun (F ↾ A) ∧ A ⊆ dom F)) |
11 | 2, 10 | sylibr 203 | 1 ⊢ ((Fun F ∧ A ⊆ dom F) → (F ↾ A):A–onto→(F “ A)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ⊆ wss 3257 “ cima 4722 dom cdm 4772 ran crn 4773 ↾ cres 4774 Fun wfun 4775 Fn wfn 4776 –onto→wfo 4779 |
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-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-fo 4793 |
This theorem is referenced by: f1ores 5300 resdif 5306 |
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