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Mirrors > Home > MPE Home > Th. List > hashimarni | Structured version Visualization version GIF version |
Description: If the size of the image of a one-to-one function πΈ under the range of a function πΉ which is a one-to-one function into the domain of πΈ is a nonnegative integer, the size of the function πΉ is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
Ref | Expression |
---|---|
hashimarni | β’ ((πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π) β ((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ π = (πΈ β ran πΉ) β§ (β―βπ) = π) β (β―βπΉ) = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6901 | . . . . . . . 8 β’ (π = (πΈ β ran πΉ) β ((β―βπ) = π β (β―β(πΈ β ran πΉ)) = π)) | |
2 | 1 | adantl 480 | . . . . . . 7 β’ (((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ (πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π)) β§ π = (πΈ β ran πΉ)) β ((β―βπ) = π β (β―β(πΈ β ran πΉ)) = π)) |
3 | hashimarn 14431 | . . . . . . . . . . 11 β’ ((πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π) β (πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β (β―β(πΈ β ran πΉ)) = (β―βπΉ))) | |
4 | 3 | impcom 406 | . . . . . . . . . 10 β’ ((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ (πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π)) β (β―β(πΈ β ran πΉ)) = (β―βπΉ)) |
5 | id 22 | . . . . . . . . . 10 β’ ((β―β(πΈ β ran πΉ)) = π β (β―β(πΈ β ran πΉ)) = π) | |
6 | 4, 5 | sylan9req 2786 | . . . . . . . . 9 β’ (((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ (πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π)) β§ (β―β(πΈ β ran πΉ)) = π) β (β―βπΉ) = π) |
7 | 6 | ex 411 | . . . . . . . 8 β’ ((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ (πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π)) β ((β―β(πΈ β ran πΉ)) = π β (β―βπΉ) = π)) |
8 | 7 | adantr 479 | . . . . . . 7 β’ (((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ (πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π)) β§ π = (πΈ β ran πΉ)) β ((β―β(πΈ β ran πΉ)) = π β (β―βπΉ) = π)) |
9 | 2, 8 | sylbid 239 | . . . . . 6 β’ (((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ (πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π)) β§ π = (πΈ β ran πΉ)) β ((β―βπ) = π β (β―βπΉ) = π)) |
10 | 9 | exp31 418 | . . . . 5 β’ (πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β ((πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π) β (π = (πΈ β ran πΉ) β ((β―βπ) = π β (β―βπΉ) = π)))) |
11 | 10 | com23 86 | . . . 4 β’ (πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β (π = (πΈ β ran πΉ) β ((πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π) β ((β―βπ) = π β (β―βπΉ) = π)))) |
12 | 11 | com34 91 | . . 3 β’ (πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β (π = (πΈ β ran πΉ) β ((β―βπ) = π β ((πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π) β (β―βπΉ) = π)))) |
13 | 12 | 3imp 1108 | . 2 β’ ((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ π = (πΈ β ran πΉ) β§ (β―βπ) = π) β ((πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π) β (β―βπΉ) = π)) |
14 | 13 | com12 32 | 1 β’ ((πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π) β ((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ π = (πΈ β ran πΉ) β§ (β―βπ) = π) β (β―βπΉ) = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 dom cdm 5672 ran crn 5673 β cima 5675 β1-1βwf1 6540 βcfv 6543 (class class class)co 7416 0cc0 11138 ..^cfzo 13659 β―chash 14321 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-hash 14322 |
This theorem is referenced by: (None) |
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