Theorem hashimarn 14363

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

Proof of Theorem hashimarn

Step	Hyp	Ref	Expression
1		f1f 6730	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐸)
2	1	frnd 6670	. . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → ran 𝐹 ⊆ dom 𝐸)
3	2	adantl 481	. . . . 5 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → ran 𝐹 ⊆ dom 𝐸)
4		ssdmres 5972	. . . . 5 ⊢ (ran 𝐹 ⊆ dom 𝐸 ↔ dom (𝐸 ↾ ran 𝐹) = ran 𝐹)
5	3, 4	sylib 218	. . . 4 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → dom (𝐸 ↾ ran 𝐹) = ran 𝐹)
6	5	fveq2d 6838	. . 3 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘dom (𝐸 ↾ ran 𝐹)) = (♯‘ran 𝐹))
7		df-ima 5637	. . . . 5 ⊢ (𝐸 “ ran 𝐹) = ran (𝐸 ↾ ran 𝐹)
8	7	fveq2i 6837	. . . 4 ⊢ (♯‘(𝐸 “ ran 𝐹)) = (♯‘ran (𝐸 ↾ ran 𝐹))
9		f1fun 6732	. . . . . . . 8 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → Fun 𝐸)
10		funres 6534	. . . . . . . . 9 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ ran 𝐹))
11	10	funfnd 6523	. . . . . . . 8 ⊢ (Fun 𝐸 → (𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹))
12	9, 11	syl 17	. . . . . . 7 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → (𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹))
13	12	ad2antrr 726	. . . . . 6 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹))
14		hashfn 14298	. . . . . 6 ⊢ ((𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹) → (♯‘(𝐸 ↾ ran 𝐹)) = (♯‘dom (𝐸 ↾ ran 𝐹)))
15	13, 14	syl 17	. . . . 5 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘(𝐸 ↾ ran 𝐹)) = (♯‘dom (𝐸 ↾ ran 𝐹)))
16		ovex 7391	. . . . . . . 8 ⊢ (0..^(♯‘𝐹)) ∈ V
17		fex 7172	. . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ (0..^(♯‘𝐹)) ∈ V) → 𝐹 ∈ V)
18	1, 16, 17	sylancl 586	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → 𝐹 ∈ V)
19		rnexg 7844	. . . . . . 7 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
20	18, 19	syl 17	. . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → ran 𝐹 ∈ V)
21		simpll 766	. . . . . . 7 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → 𝐸:dom 𝐸–1-1→ran 𝐸)
22		f1ssres 6737	. . . . . . 7 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ ran 𝐹 ⊆ dom 𝐸) → (𝐸 ↾ ran 𝐹):ran 𝐹–1-1→ran 𝐸)
23	21, 3, 22	syl2anc 584	. . . . . 6 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (𝐸 ↾ ran 𝐹):ran 𝐹–1-1→ran 𝐸)
24		hashf1rn 14275	. . . . . 6 ⊢ ((ran 𝐹 ∈ V ∧ (𝐸 ↾ ran 𝐹):ran 𝐹–1-1→ran 𝐸) → (♯‘(𝐸 ↾ ran 𝐹)) = (♯‘ran (𝐸 ↾ ran 𝐹)))
25	20, 23, 24	syl2an2 686	. . . . 5 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘(𝐸 ↾ ran 𝐹)) = (♯‘ran (𝐸 ↾ ran 𝐹)))
26	15, 25	eqtr3d 2773	. . . 4 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘dom (𝐸 ↾ ran 𝐹)) = (♯‘ran (𝐸 ↾ ran 𝐹)))
27	8, 26	eqtr4id 2790	. . 3 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘(𝐸 “ ran 𝐹)) = (♯‘dom (𝐸 ↾ ran 𝐹)))
28		hashf1rn 14275	. . . . 5 ⊢ (((0..^(♯‘𝐹)) ∈ V ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘𝐹) = (♯‘ran 𝐹))
29	16, 28	mpan 690	. . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (♯‘𝐹) = (♯‘ran 𝐹))
30	29	adantl 481	. . 3 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘𝐹) = (♯‘ran 𝐹))
31	6, 27, 30	3eqtr4d 2781	. 2 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹))
32	31	ex 412	1 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  dom cdm 5624  ran crn 5625   ↾ cres 5626   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  (class class class)co 7358  0cc0 11026  ..^cfzo 13570  ♯chash 14253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-hash 14254

This theorem is referenced by:  hashimarni  14364