Theorem hashimarn 14461

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 6784	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐸)
2	1	frnd 6724	. . . . . 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 6011	. . . . 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 6890	. . 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 5678	. . . . 5 ⊢ (𝐸 “ ran 𝐹) = ran (𝐸 ↾ ran 𝐹)
8	7	fveq2i 6889	. . . 4 ⊢ (♯‘(𝐸 “ ran 𝐹)) = (♯‘ran (𝐸 ↾ ran 𝐹))
9		f1fun 6786	. . . . . . . 8 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → Fun 𝐸)
10		funres 6588	. . . . . . . . 9 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ ran 𝐹))
11	10	funfnd 6577	. . . . . . . 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 14396	. . . . . 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 7446	. . . . . . . 8 ⊢ (0..^(♯‘𝐹)) ∈ V
17		fex 7228	. . . . . . . 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 7906	. . . . . . 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 6791	. . . . . . 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 14373	. . . . . 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 2771	. . . 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 2788	. . 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 14373	. . . . 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 2779	. 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 1539   ∈ wcel 2107  Vcvv 3463   ⊆ wss 3931  dom cdm 5665  ran crn 5666   ↾ cres 5667   “ cima 5668  Fun wfun 6535   Fn wfn 6536  ⟶wf 6537  –1-1→wf1 6538  ‘cfv 6541  (class class class)co 7413  0cc0 11137  ..^cfzo 13676  ♯chash 14351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-card 9961  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-n0 12510  df-z 12597  df-uz 12861  df-hash 14352

This theorem is referenced by:  hashimarni  14462