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Theorem hashimarni 13266
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 𝐸1-1→ran 𝐸𝐸𝑉) → ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃 = (𝐸 “ ran 𝐹) ∧ (#‘𝑃) = 𝑁) → (#‘𝐹) = 𝑁))

Proof of Theorem hashimarni
StepHypRef Expression
1 fveq2 6229 . . . . . . . . 9 (𝑃 = (𝐸 “ ran 𝐹) → (#‘𝑃) = (#‘(𝐸 “ ran 𝐹)))
21eqeq1d 2653 . . . . . . . 8 (𝑃 = (𝐸 “ ran 𝐹) → ((#‘𝑃) = 𝑁 ↔ (#‘(𝐸 “ ran 𝐹)) = 𝑁))
32adantl 481 . . . . . . 7 (((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉)) ∧ 𝑃 = (𝐸 “ ran 𝐹)) → ((#‘𝑃) = 𝑁 ↔ (#‘(𝐸 “ ran 𝐹)) = 𝑁))
4 hashimarn 13265 . . . . . . . . . . 11 ((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → (#‘(𝐸 “ ran 𝐹)) = (#‘𝐹)))
54impcom 445 . . . . . . . . . 10 ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉)) → (#‘(𝐸 “ ran 𝐹)) = (#‘𝐹))
6 id 22 . . . . . . . . . 10 ((#‘(𝐸 “ ran 𝐹)) = 𝑁 → (#‘(𝐸 “ ran 𝐹)) = 𝑁)
75, 6sylan9req 2706 . . . . . . . . 9 (((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉)) ∧ (#‘(𝐸 “ ran 𝐹)) = 𝑁) → (#‘𝐹) = 𝑁)
87ex 449 . . . . . . . 8 ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉)) → ((#‘(𝐸 “ ran 𝐹)) = 𝑁 → (#‘𝐹) = 𝑁))
98adantr 480 . . . . . . 7 (((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉)) ∧ 𝑃 = (𝐸 “ ran 𝐹)) → ((#‘(𝐸 “ ran 𝐹)) = 𝑁 → (#‘𝐹) = 𝑁))
103, 9sylbid 230 . . . . . 6 (((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉)) ∧ 𝑃 = (𝐸 “ ran 𝐹)) → ((#‘𝑃) = 𝑁 → (#‘𝐹) = 𝑁))
1110exp31 629 . . . . 5 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → ((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → (𝑃 = (𝐸 “ ran 𝐹) → ((#‘𝑃) = 𝑁 → (#‘𝐹) = 𝑁))))
1211com23 86 . . . 4 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → (𝑃 = (𝐸 “ ran 𝐹) → ((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → ((#‘𝑃) = 𝑁 → (#‘𝐹) = 𝑁))))
1312com34 91 . . 3 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → (𝑃 = (𝐸 “ ran 𝐹) → ((#‘𝑃) = 𝑁 → ((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → (#‘𝐹) = 𝑁))))
14133imp 1275 . 2 ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃 = (𝐸 “ ran 𝐹) ∧ (#‘𝑃) = 𝑁) → ((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → (#‘𝐹) = 𝑁))
1514com12 32 1 ((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃 = (𝐸 “ ran 𝐹) ∧ (#‘𝑃) = 𝑁) → (#‘𝐹) = 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  dom cdm 5143  ran crn 5144  cima 5146  1-1wf1 5923  cfv 5926  (class class class)co 6690  0cc0 9974  ..^cfzo 12504  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-hash 13158
This theorem is referenced by: (None)
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