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Mirrors  >  Home  >  MPE Home  >  Th. List  >  hashimarni Structured version   Visualization version   GIF version
Theorem hashimarni 14432
Description: If the size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 is a nonnegative integer, the size of the function 𝐹 is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
Assertion
Ref Expression
hashimarni ((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 ∧ 𝑃 = (𝐸 β€œ ran 𝐹) ∧ (β™―β€˜π‘ƒ) = 𝑁) β†’ (β™―β€˜πΉ) = 𝑁))
Proof of Theorem hashimarni
StepHypRef Expression
1 fveqeq2 6901 . . . . . . . 8 (𝑃 = (𝐸 β€œ ran 𝐹) β†’ ((β™―β€˜π‘ƒ) = 𝑁 ↔ (β™―β€˜(𝐸 β€œ ran 𝐹)) = 𝑁))
21adantl 480 . . . . . . 7 (((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 ∧ (𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ 𝑃 = (𝐸 β€œ ran 𝐹)) β†’ ((β™―β€˜π‘ƒ) = 𝑁 ↔ (β™―β€˜(𝐸 β€œ ran 𝐹)) = 𝑁))
3 hashimarn 14431 . . . . . . . . . . 11 ((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 β†’ (β™―β€˜(𝐸 β€œ ran 𝐹)) = (β™―β€˜πΉ)))
43impcom 406 . . . . . . . . . 10 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 ∧ (𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉)) β†’ (β™―β€˜(𝐸 β€œ ran 𝐹)) = (β™―β€˜πΉ))
5 id 22 . . . . . . . . . 10 ((β™―β€˜(𝐸 β€œ ran 𝐹)) = 𝑁 β†’ (β™―β€˜(𝐸 β€œ ran 𝐹)) = 𝑁)
64, 5sylan9req 2786 . . . . . . . . 9 (((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 ∧ (𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ (β™―β€˜(𝐸 β€œ ran 𝐹)) = 𝑁) β†’ (β™―β€˜πΉ) = 𝑁)
76ex 411 . . . . . . . 8 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 ∧ (𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉)) β†’ ((β™―β€˜(𝐸 β€œ ran 𝐹)) = 𝑁 β†’ (β™―β€˜πΉ) = 𝑁))
87adantr 479 . . . . . . 7 (((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 ∧ (𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ 𝑃 = (𝐸 β€œ ran 𝐹)) β†’ ((β™―β€˜(𝐸 β€œ ran 𝐹)) = 𝑁 β†’ (β™―β€˜πΉ) = 𝑁))
92, 8sylbid 239 . . . . . 6 (((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 ∧ (𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ 𝑃 = (𝐸 β€œ ran 𝐹)) β†’ ((β™―β€˜π‘ƒ) = 𝑁 β†’ (β™―β€˜πΉ) = 𝑁))
109exp31 418 . . . . 5 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 β†’ ((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) β†’ (𝑃 = (𝐸 β€œ ran 𝐹) β†’ ((β™―β€˜π‘ƒ) = 𝑁 β†’ (β™―β€˜πΉ) = 𝑁))))
1110com23 86 . . . 4 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 β†’ (𝑃 = (𝐸 β€œ ran 𝐹) β†’ ((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) β†’ ((β™―β€˜π‘ƒ) = 𝑁 β†’ (β™―β€˜πΉ) = 𝑁))))
1211com34 91 . . 3 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 β†’ (𝑃 = (𝐸 β€œ ran 𝐹) β†’ ((β™―β€˜π‘ƒ) = 𝑁 β†’ ((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) β†’ (β™―β€˜πΉ) = 𝑁))))
13123imp 1108 . 2 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 ∧ 𝑃 = (𝐸 β€œ ran 𝐹) ∧ (β™―β€˜π‘ƒ) = 𝑁) β†’ ((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) β†’ (β™―β€˜πΉ) = 𝑁))
1413com12 32 1 ((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 ∧ 𝑃 = (𝐸 β€œ ran 𝐹) ∧ (β™―β€˜π‘ƒ) = 𝑁) β†’ (β™―β€˜πΉ) = 𝑁))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  dom cdm 5672  ran crn 5673   β€œ cima 5675  β€“1-1β†’wf1 6540  β€˜cfv 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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