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Mirrors > Home > NFE Home > Th. List > ffoss | GIF version |
Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by set.mm contributors, 10-May-1998.) |
Ref | Expression |
---|---|
f11o.1 | ⊢ F ∈ V |
Ref | Expression |
---|---|
ffoss | ⊢ (F:A–→B ↔ ∃x(F:A–onto→x ∧ x ⊆ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 4791 | . . . 4 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B)) | |
2 | dffn4 5275 | . . . . 5 ⊢ (F Fn A ↔ F:A–onto→ran F) | |
3 | 2 | anbi1i 676 | . . . 4 ⊢ ((F Fn A ∧ ran F ⊆ B) ↔ (F:A–onto→ran F ∧ ran F ⊆ B)) |
4 | 1, 3 | bitri 240 | . . 3 ⊢ (F:A–→B ↔ (F:A–onto→ran F ∧ ran F ⊆ B)) |
5 | f11o.1 | . . . . 5 ⊢ F ∈ V | |
6 | 5 | rnex 5107 | . . . 4 ⊢ ran F ∈ V |
7 | foeq3 5267 | . . . . 5 ⊢ (x = ran F → (F:A–onto→x ↔ F:A–onto→ran F)) | |
8 | sseq1 3292 | . . . . 5 ⊢ (x = ran F → (x ⊆ B ↔ ran F ⊆ B)) | |
9 | 7, 8 | anbi12d 691 | . . . 4 ⊢ (x = ran F → ((F:A–onto→x ∧ x ⊆ B) ↔ (F:A–onto→ran F ∧ ran F ⊆ B))) |
10 | 6, 9 | spcev 2946 | . . 3 ⊢ ((F:A–onto→ran F ∧ ran F ⊆ B) → ∃x(F:A–onto→x ∧ x ⊆ B)) |
11 | 4, 10 | sylbi 187 | . 2 ⊢ (F:A–→B → ∃x(F:A–onto→x ∧ x ⊆ B)) |
12 | fof 5269 | . . . 4 ⊢ (F:A–onto→x → F:A–→x) | |
13 | fss 5230 | . . . 4 ⊢ ((F:A–→x ∧ x ⊆ B) → F:A–→B) | |
14 | 12, 13 | sylan 457 | . . 3 ⊢ ((F:A–onto→x ∧ x ⊆ B) → F:A–→B) |
15 | 14 | exlimiv 1634 | . 2 ⊢ (∃x(F:A–onto→x ∧ x ⊆ B) → F:A–→B) |
16 | 11, 15 | impbii 180 | 1 ⊢ (F:A–→B ↔ ∃x(F:A–onto→x ∧ x ⊆ B)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ⊆ wss 3257 ran crn 4773 Fn wfn 4776 –→wf 4777 –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-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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-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-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-br 4640 df-ima 4727 df-rn 4786 df-f 4791 df-fo 4793 |
This theorem is referenced by: f11o 5315 |
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