MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hashimarni Structured version   Visualization version   GIF version
Theorem hashimarni 14406
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 6894 . . . . . . . 8 (𝑃 = (𝐸 β€œ ran 𝐹) β†’ ((β™―β€˜π‘ƒ) = 𝑁 ↔ (β™―β€˜(𝐸 β€œ ran 𝐹)) = 𝑁))
21adantl 481 . . . . . . 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 14405 . . . . . . . . . . 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 407 . . . . . . . . . 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 2787 . . . . . . . . 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 412 . . . . . . . 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 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€˜(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 419 . . . . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  dom cdm 5669  ran crn 5670   β€œ cima 5672  β€“1-1β†’wf1 6534  β€˜cfv 6537  (class class class)co 7405  0cc0 11112  ..^cfzo 13633  β™―chash 14295
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-hash 14296
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator