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Theorem hashimarn 14347
Description: 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 equals the size of the function 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) (Proof shortened by AV, 4-May-2021.)
Assertion
Ref Expression
hashimarn ((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)))

Proof of Theorem hashimarn
StepHypRef Expression
1 f1f 6743 . . . . . . 7 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸)
21frnd 6681 . . . . . 6 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 β†’ ran 𝐹 βŠ† dom 𝐸)
32adantl 483 . . . . 5 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ ran 𝐹 βŠ† dom 𝐸)
4 ssdmres 5965 . . . . 5 (ran 𝐹 βŠ† dom 𝐸 ↔ dom (𝐸 β†Ύ ran 𝐹) = ran 𝐹)
53, 4sylib 217 . . . 4 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ dom (𝐸 β†Ύ ran 𝐹) = ran 𝐹)
65fveq2d 6851 . . 3 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (β™―β€˜dom (𝐸 β†Ύ ran 𝐹)) = (β™―β€˜ran 𝐹))
7 df-ima 5651 . . . . 5 (𝐸 β€œ ran 𝐹) = ran (𝐸 β†Ύ ran 𝐹)
87fveq2i 6850 . . . 4 (β™―β€˜(𝐸 β€œ ran 𝐹)) = (β™―β€˜ran (𝐸 β†Ύ ran 𝐹))
9 f1fun 6745 . . . . . . . 8 (𝐸:dom 𝐸–1-1β†’ran 𝐸 β†’ Fun 𝐸)
10 funres 6548 . . . . . . . . 9 (Fun 𝐸 β†’ Fun (𝐸 β†Ύ ran 𝐹))
1110funfnd 6537 . . . . . . . 8 (Fun 𝐸 β†’ (𝐸 β†Ύ ran 𝐹) Fn dom (𝐸 β†Ύ ran 𝐹))
129, 11syl 17 . . . . . . 7 (𝐸:dom 𝐸–1-1β†’ran 𝐸 β†’ (𝐸 β†Ύ ran 𝐹) Fn dom (𝐸 β†Ύ ran 𝐹))
1312ad2antrr 725 . . . . . 6 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (𝐸 β†Ύ ran 𝐹) Fn dom (𝐸 β†Ύ ran 𝐹))
14 hashfn 14282 . . . . . 6 ((𝐸 β†Ύ ran 𝐹) Fn dom (𝐸 β†Ύ ran 𝐹) β†’ (β™―β€˜(𝐸 β†Ύ ran 𝐹)) = (β™―β€˜dom (𝐸 β†Ύ ran 𝐹)))
1513, 14syl 17 . . . . 5 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (β™―β€˜(𝐸 β†Ύ ran 𝐹)) = (β™―β€˜dom (𝐸 β†Ύ ran 𝐹)))
16 ovex 7395 . . . . . . . 8 (0..^(β™―β€˜πΉ)) ∈ V
17 fex 7181 . . . . . . . 8 ((𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸 ∧ (0..^(β™―β€˜πΉ)) ∈ V) β†’ 𝐹 ∈ V)
181, 16, 17sylancl 587 . . . . . . 7 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 β†’ 𝐹 ∈ V)
19 rnexg 7846 . . . . . . 7 (𝐹 ∈ V β†’ ran 𝐹 ∈ V)
2018, 19syl 17 . . . . . 6 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 β†’ ran 𝐹 ∈ V)
21 simpll 766 . . . . . . 7 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ 𝐸:dom 𝐸–1-1β†’ran 𝐸)
22 f1ssres 6751 . . . . . . 7 ((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ ran 𝐹 βŠ† dom 𝐸) β†’ (𝐸 β†Ύ ran 𝐹):ran 𝐹–1-1β†’ran 𝐸)
2321, 3, 22syl2anc 585 . . . . . 6 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (𝐸 β†Ύ ran 𝐹):ran 𝐹–1-1β†’ran 𝐸)
24 hashf1rn 14259 . . . . . 6 ((ran 𝐹 ∈ V ∧ (𝐸 β†Ύ ran 𝐹):ran 𝐹–1-1β†’ran 𝐸) β†’ (β™―β€˜(𝐸 β†Ύ ran 𝐹)) = (β™―β€˜ran (𝐸 β†Ύ ran 𝐹)))
2520, 23, 24syl2an2 685 . . . . 5 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (β™―β€˜(𝐸 β†Ύ ran 𝐹)) = (β™―β€˜ran (𝐸 β†Ύ ran 𝐹)))
2615, 25eqtr3d 2779 . . . 4 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (β™―β€˜dom (𝐸 β†Ύ ran 𝐹)) = (β™―β€˜ran (𝐸 β†Ύ ran 𝐹)))
278, 26eqtr4id 2796 . . 3 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (β™―β€˜(𝐸 β€œ ran 𝐹)) = (β™―β€˜dom (𝐸 β†Ύ ran 𝐹)))
28 hashf1rn 14259 . . . . 5 (((0..^(β™―β€˜πΉ)) ∈ V ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (β™―β€˜πΉ) = (β™―β€˜ran 𝐹))
2916, 28mpan 689 . . . 4 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸 β†’ (β™―β€˜πΉ) = (β™―β€˜ran 𝐹))
3029adantl 483 . . 3 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (β™―β€˜πΉ) = (β™―β€˜ran 𝐹))
316, 27, 303eqtr4d 2787 . 2 (((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸) β†’ (β™―β€˜(𝐸 β€œ ran 𝐹)) = (β™―β€˜πΉ))
3231ex 414 1 ((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)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3448   βŠ† wss 3915  dom cdm 5638  ran crn 5639   β†Ύ cres 5640   β€œ cima 5641  Fun wfun 6495   Fn wfn 6496  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€˜cfv 6501  (class class class)co 7362  0cc0 11058  ..^cfzo 13574  β™―chash 14237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-hash 14238
This theorem is referenced by:  hashimarni  14348
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