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Mirrors > Home > MPE Home > Th. List > intrnfi | Structured version Visualization version GIF version |
Description: Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
intrnfi | β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β β© ran πΉ β (fiβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1191 | . . . 4 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β πΉ:π΄βΆπ΅) | |
2 | 1 | frnd 6735 | . . 3 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β ran πΉ β π΅) |
3 | 1 | fdmd 6738 | . . . . 5 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β dom πΉ = π΄) |
4 | simpr2 1192 | . . . . 5 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β π΄ β β ) | |
5 | 3, 4 | eqnetrd 3005 | . . . 4 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β dom πΉ β β ) |
6 | dm0rn0 5931 | . . . . 5 β’ (dom πΉ = β β ran πΉ = β ) | |
7 | 6 | necon3bii 2990 | . . . 4 β’ (dom πΉ β β β ran πΉ β β ) |
8 | 5, 7 | sylib 217 | . . 3 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β ran πΉ β β ) |
9 | simpr3 1193 | . . . 4 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β π΄ β Fin) | |
10 | 1 | ffnd 6728 | . . . . 5 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β πΉ Fn π΄) |
11 | dffn4 6822 | . . . . 5 β’ (πΉ Fn π΄ β πΉ:π΄βontoβran πΉ) | |
12 | 10, 11 | sylib 217 | . . . 4 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β πΉ:π΄βontoβran πΉ) |
13 | fofi 9370 | . . . 4 β’ ((π΄ β Fin β§ πΉ:π΄βontoβran πΉ) β ran πΉ β Fin) | |
14 | 9, 12, 13 | syl2anc 582 | . . 3 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β ran πΉ β Fin) |
15 | 2, 8, 14 | 3jca 1125 | . 2 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β (ran πΉ β π΅ β§ ran πΉ β β β§ ran πΉ β Fin)) |
16 | elfir 9446 | . 2 β’ ((π΅ β π β§ (ran πΉ β π΅ β§ ran πΉ β β β§ ran πΉ β Fin)) β β© ran πΉ β (fiβπ΅)) | |
17 | 15, 16 | syldan 589 | 1 β’ ((π΅ β π β§ (πΉ:π΄βΆπ΅ β§ π΄ β β β§ π΄ β Fin)) β β© ran πΉ β (fiβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 β wcel 2098 β wne 2937 β wss 3949 β c0 4326 β© cint 4953 dom cdm 5682 ran crn 5683 Fn wfn 6548 βΆwf 6549 βontoβwfo 6551 βcfv 6553 Fincfn 8970 ficfi 9441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-fin 8974 df-fi 9442 |
This theorem is referenced by: iinfi 9448 firest 17421 |
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