Theorem hashimarn 14565

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 6770	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐸)
2	1	frnd 6710	. . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → ran 𝐹 ⊆ dom 𝐸)
3	2	adantl 487	. . . . 5 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → ran 𝐹 ⊆ dom 𝐸)
4		ssdmres 6004	. . . . 5 ⊢ (ran 𝐹 ⊆ dom 𝐸 ↔ dom (𝐸 ↾ ran 𝐹) = ran 𝐹)
5	3, 4	sylib 221	. . . 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 6881	. . 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 5664	. . . . 5 ⊢ (𝐸 “ ran 𝐹) = ran (𝐸 ↾ ran 𝐹)
8	7	fveq2i 6880	. . . 4 ⊢ (♯‘(𝐸 “ ran 𝐹)) = (♯‘ran (𝐸 ↾ ran 𝐹))
9		f1fun 6772	. . . . . . . 8 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → Fun 𝐸)
10		funres 6574	. . . . . . . . 9 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ ran 𝐹))
11	10	funfnd 6563	. . . . . . . 8 ⊢ (Fun 𝐸 → (𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹))
12	9, 11	syl 18	. . . . . . 7 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → (𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹))
13	12	ad2antrr 739	. . . . . 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 14499	. . . . . 6 ⊢ ((𝐸 ↾ ran 𝐹) Fn dom (𝐸 ↾ ran 𝐹) → (♯‘(𝐸 ↾ ran 𝐹)) = (♯‘dom (𝐸 ↾ ran 𝐹)))
15	13, 14	syl 18	. . . . 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 7445	. . . . . . . 8 ⊢ (0..^(♯‘𝐹)) ∈ V
17		fex 7224	. . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ (0..^(♯‘𝐹)) ∈ V) → 𝐹 ∈ V)
18	1, 16, 17	sylancl 598	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → 𝐹 ∈ V)
19		rnexg 7903	. . . . . . 7 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
20	18, 19	syl 18	. . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → ran 𝐹 ∈ V)
21		simpll 779	. . . . . . 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 6779	. . . . . . 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 596	. . . . . 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 14476	. . . . . 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 699	. . . . 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 2798	. . . 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 2815	. . 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 14476	. . . . 5 ⊢ (((0..^(♯‘𝐹)) ∈ V ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘𝐹) = (♯‘ran 𝐹))
29	16, 28	mpan 703	. . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (♯‘𝐹) = (♯‘ran 𝐹))
30	29	adantl 487	. . 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 2806	. 2 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸) → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹))
32	31	ex 418	1 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527  –1-1→wf1 6528  ‘cfv 6531  (class class class)co 7412  0cc0 11181  ..^cfzo 13768  ♯chash 14454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-n0 12588  df-z 12675  df-uz 12947  df-hash 14455

This theorem is used by:  hashimarni  14566