Theorem hashimarn 13085

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 6014	. . . . . . 7 ⊢ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
2		frn 5966	. . . . . . 7 ⊢ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 → ran 𝐹 ⊆ dom 𝐸)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → ran 𝐹 ⊆ dom 𝐸)
4	3	adantl 481	. . . . 5 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) → ran 𝐹 ⊆ dom 𝐸)
5		ssdmres 5340	. . . . 5 ⊢ (ran 𝐹 ⊆ dom 𝐸 ↔ dom (𝐸 ↾ ran 𝐹) = ran 𝐹)
6	4, 5	sylib 207	. . . 4 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) → dom (𝐸 ↾ ran 𝐹) = ran 𝐹)
7	6	fveq2d 6107	. . 3 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) → (#‘dom (𝐸 ↾ ran 𝐹)) = (#‘ran 𝐹))
8		f1fun 6016	. . . . . . . 8 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → Fun 𝐸)
9		funres 5843	. . . . . . . . 9 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ ran 𝐹))
10		funfn 5833	. . . . . . . . 9 ⊢ (Fun (𝐸 ↾ ran 𝐹) ↔ (𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹))
11	9, 10	sylib 207	. . . . . . . 8 ⊢ (Fun 𝐸 → (𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹))
12	8, 11	syl 17	. . . . . . 7 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → (𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹))
13	12	ad2antrr 758	. . . . . 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 13025	. . . . . 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 6577	. . . . . . . 8 ⊢ (0..^(#‘𝐹)) ∈ V
17		fex 6394	. . . . . . . 8 ⊢ ((𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ (0..^(#‘𝐹)) ∈ V) → 𝐹 ∈ V)
18	1, 16, 17	sylancl 693	. . . . . . 7 ⊢ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → 𝐹 ∈ V)
19		rnexg 6990	. . . . . . 7 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
20	18, 19	syl 17	. . . . . 6 ⊢ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → ran 𝐹 ∈ V)
21		simpll 786	. . . . . . 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 6021	. . . . . . 7 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ ran 𝐹 ⊆ dom 𝐸) → (𝐸 ↾ ran 𝐹):ran 𝐹–1-1→ran 𝐸)
23	21, 4, 22	syl2anc 691	. . . . . 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 13004	. . . . . 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 871	. . . . 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 2646	. . . 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		df-ima 5051	. . . . 5 ⊢ (𝐸 “ ran 𝐹) = ran (𝐸 ↾ ran 𝐹)
28	27	fveq2i 6106	. . . 4 ⊢ (#‘(𝐸 “ ran 𝐹)) = (#‘ran (𝐸 ↾ ran 𝐹))
29	26, 28	syl6reqr 2663	. . 3 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) → (#‘(𝐸 “ ran 𝐹)) = (#‘dom (𝐸 ↾ ran 𝐹)))
30		hashf1rn 13004	. . . . 5 ⊢ (((0..^(#‘𝐹)) ∈ V ∧ 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) → (#‘𝐹) = (#‘ran 𝐹))
31	16, 30	mpan 702	. . . 4 ⊢ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → (#‘𝐹) = (#‘ran 𝐹))
32	31	adantl 481	. . 3 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) → (#‘𝐹) = (#‘ran 𝐹))
33	7, 29, 32	3eqtr4d 2654	. 2 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) → (#‘(𝐸 “ ran 𝐹)) = (#‘𝐹))
34	33	ex 449	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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ..^cfzo 12334  #chash 12979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-hash 12980

This theorem is referenced by:  hashimarni  13086