Theorem hashimarni 14084

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–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ 𝑃 = (𝐸 “ ran 𝐹) ∧ (♯‘𝑃) = 𝑁) → (♯‘𝐹) = 𝑁))

Proof of Theorem hashimarni

Step	Hyp	Ref	Expression
1		fveqeq2 6765	. . . . . . . 8 ⊢ (𝑃 = (𝐸 “ ran 𝐹) → ((♯‘𝑃) = 𝑁 ↔ (♯‘(𝐸 “ ran 𝐹)) = 𝑁))
2	1	adantl 481	. . . . . . 7 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ 𝑃 = (𝐸 “ ran 𝐹)) → ((♯‘𝑃) = 𝑁 ↔ (♯‘(𝐸 “ ran 𝐹)) = 𝑁))
3		hashimarn 14083	. . . . . . . . . . 11 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹)))
4	3	impcom 407	. . . . . . . . . 10 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹))
5		id 22	. . . . . . . . . 10 ⊢ ((♯‘(𝐸 “ ran 𝐹)) = 𝑁 → (♯‘(𝐸 “ ran 𝐹)) = 𝑁)
6	4, 5	sylan9req 2800	. . . . . . . . 9 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ (♯‘(𝐸 “ ran 𝐹)) = 𝑁) → (♯‘𝐹) = 𝑁)
7	6	ex 412	. . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) → ((♯‘(𝐸 “ ran 𝐹)) = 𝑁 → (♯‘𝐹) = 𝑁))
8	7	adantr 480	. . . . . . 7 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ 𝑃 = (𝐸 “ ran 𝐹)) → ((♯‘(𝐸 “ ran 𝐹)) = 𝑁 → (♯‘𝐹) = 𝑁))
9	2, 8	sylbid 239	. . . . . 6 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ 𝑃 = (𝐸 “ ran 𝐹)) → ((♯‘𝑃) = 𝑁 → (♯‘𝐹) = 𝑁))
10	9	exp31 419	. . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (𝑃 = (𝐸 “ ran 𝐹) → ((♯‘𝑃) = 𝑁 → (♯‘𝐹) = 𝑁))))
11	10	com23 86	. . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (𝑃 = (𝐸 “ ran 𝐹) → ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → ((♯‘𝑃) = 𝑁 → (♯‘𝐹) = 𝑁))))
12	11	com34 91	. . 3 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (𝑃 = (𝐸 “ ran 𝐹) → ((♯‘𝑃) = 𝑁 → ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (♯‘𝐹) = 𝑁))))
13	12	3imp 1109	. 2 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ 𝑃 = (𝐸 “ ran 𝐹) ∧ (♯‘𝑃) = 𝑁) → ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (♯‘𝐹) = 𝑁))
14	13	com12 32	1 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ 𝑃 = (𝐸 “ ran 𝐹) ∧ (♯‘𝑃) = 𝑁) → (♯‘𝐹) = 𝑁))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  dom cdm 5580  ran crn 5581   “ cima 5583  –1-1→wf1 6415  ‘cfv 6418  (class class class)co 7255  0cc0 10802  ..^cfzo 13311  ♯chash 13972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-hash 13973

This theorem is referenced by: (None)