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Mirrors  >  Home  >  MPE Home  >  Th. List  >  hashimarni Structured version   Visualization version   GIF version
Theorem hashimarni 14400
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 6900 . . . . . . . 8 (𝑃 = (𝐸 β€œ ran 𝐹) β†’ ((β™―β€˜π‘ƒ) = 𝑁 ↔ (β™―β€˜(𝐸 β€œ ran 𝐹)) = 𝑁))
21adantl 482 . . . . . . 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 14399 . . . . . . . . . . 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 408 . . . . . . . . . 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 2793 . . . . . . . . 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 413 . . . . . . . 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 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€˜(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 420 . . . . 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 1111 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  dom cdm 5676  ran crn 5677   β€œ cima 5679  β€“1-1β†’wf1 6540  β€˜cfv 6543  (class class class)co 7408  0cc0 11109  ..^cfzo 13626  β™―chash 14289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-hash 14290
This theorem is referenced by: (None)
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