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Mirrors > Home > NFE Home > Th. List > fint | GIF version |
Description: Function into an intersection. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Oct-1999.) (Revised by set.mm contributors, 18-Sep-2011.) |
Ref | Expression |
---|---|
fint.1 | ⊢ B ≠ ∅ |
Ref | Expression |
---|---|
fint | ⊢ (F:A–→∩B ↔ ∀x ∈ B F:A–→x) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 3942 | . . . 4 ⊢ (ran F ⊆ ∩B ↔ ∀x ∈ B ran F ⊆ x) | |
2 | 1 | anbi2i 675 | . . 3 ⊢ ((F Fn A ∧ ran F ⊆ ∩B) ↔ (F Fn A ∧ ∀x ∈ B ran F ⊆ x)) |
3 | fint.1 | . . . 4 ⊢ B ≠ ∅ | |
4 | r19.28zv 3645 | . . . 4 ⊢ (B ≠ ∅ → (∀x ∈ B (F Fn A ∧ ran F ⊆ x) ↔ (F Fn A ∧ ∀x ∈ B ran F ⊆ x))) | |
5 | 3, 4 | ax-mp 8 | . . 3 ⊢ (∀x ∈ B (F Fn A ∧ ran F ⊆ x) ↔ (F Fn A ∧ ∀x ∈ B ran F ⊆ x)) |
6 | 2, 5 | bitr4i 243 | . 2 ⊢ ((F Fn A ∧ ran F ⊆ ∩B) ↔ ∀x ∈ B (F Fn A ∧ ran F ⊆ x)) |
7 | df-f 4791 | . 2 ⊢ (F:A–→∩B ↔ (F Fn A ∧ ran F ⊆ ∩B)) | |
8 | df-f 4791 | . . 3 ⊢ (F:A–→x ↔ (F Fn A ∧ ran F ⊆ x)) | |
9 | 8 | ralbii 2638 | . 2 ⊢ (∀x ∈ B F:A–→x ↔ ∀x ∈ B (F Fn A ∧ ran F ⊆ x)) |
10 | 6, 7, 9 | 3bitr4i 268 | 1 ⊢ (F:A–→∩B ↔ ∀x ∈ B F:A–→x) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ≠ wne 2516 ∀wral 2614 ⊆ wss 3257 ∅c0 3550 ∩cint 3926 ran crn 4773 Fn wfn 4776 –→wf 4777 |
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 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 df-int 3927 df-f 4791 |
This theorem is referenced by: (None) |
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