Theorem hashimarn 14458

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 6774	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐸)
2	1	frnd 6714	. . . . . 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 6000	. . . . 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 6880	. . 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 5667	. . . . 5 ⊢ (𝐸 “ ran 𝐹) = ran (𝐸 ↾ ran 𝐹)
8	7	fveq2i 6879	. . . 4 ⊢ (♯‘(𝐸 “ ran 𝐹)) = (♯‘ran (𝐸 ↾ ran 𝐹))
9		f1fun 6776	. . . . . . . 8 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → Fun 𝐸)
10		funres 6578	. . . . . . . . 9 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ ran 𝐹))
11	10	funfnd 6567	. . . . . . . 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 14393	. . . . . 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 7438	. . . . . . . 8 ⊢ (0..^(♯‘𝐹)) ∈ V
17		fex 7218	. . . . . . . 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 7898	. . . . . . 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 6781	. . . . . . 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 14370	. . . . . 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 2772	. . . 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 2789	. . 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 14370	. . . . 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 2780	. 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 1540   ∈ wcel 2108  Vcvv 3459   ⊆ wss 3926  dom cdm 5654  ran crn 5655   ↾ cres 5656   “ cima 5657  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527  –1-1→wf1 6528  ‘cfv 6531  (class class class)co 7405  0cc0 11129  ..^cfzo 13671  ♯chash 14348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853  df-hash 14349

This theorem is referenced by:  hashimarni  14459