Theorem hasheqf1oi 14390

Description: The size of two sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 25-Dec-2017.) (Revised by AV, 4-May-2021.)

Assertion

Ref	Expression
hasheqf1oi	⊢ (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝑉

Proof of Theorem hasheqf1oi

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hasheqf1o 14388	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
2	1	biimprd 248	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))
3	2	a1d 25	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))))
4		fiinfnf1o 14389	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
5	4	pm2.21d 121	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))
6	5	a1d 25	. 2 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))))
7		fiinfnf1o 14389	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ ¬ 𝐴 ∈ Fin) → ¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)
8		19.41v 1949	. . . . . . 7 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉))
9		f1orel 6851	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → Rel 𝑓)
10	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → Rel 𝑓)
11		f1ocnvb 6861	. . . . . . . . . . . 12 ⊢ (Rel 𝑓 → (𝑓:𝐴–1-1-onto→𝐵 ↔ ◡𝑓:𝐵–1-1-onto→𝐴))
12	10, 11	syl 17	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑓:𝐴–1-1-onto→𝐵 ↔ ◡𝑓:𝐵–1-1-onto→𝐴))
13		f1of 6848	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
14		fex 7246	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝑓 ∈ V)
15	13, 14	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → 𝑓 ∈ V)
16		cnvexg 7946	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ V → ◡𝑓 ∈ V)
17		f1oeq1 6836	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵–1-1-onto→𝐴 ↔ ◡𝑓:𝐵–1-1-onto→𝐴))
18	17	spcegv 3597	. . . . . . . . . . . . 13 ⊢ (◡𝑓 ∈ V → (◡𝑓:𝐵–1-1-onto→𝐴 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴))
19	15, 16, 18	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (◡𝑓:𝐵–1-1-onto→𝐴 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴))
20		pm2.24 124	. . . . . . . . . . . 12 ⊢ (∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵)))
21	19, 20	syl6 35	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (◡𝑓:𝐵–1-1-onto→𝐴 → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵))))
22	12, 21	sylbid 240	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑓:𝐴–1-1-onto→𝐵 → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵))))
23	22	com12 32	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵))))
24	23	anabsi5 669	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵)))
25	24	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵)))
26	8, 25	sylbir 235	. . . . . 6 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵)))
27	26	ex 412	. . . . 5 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ 𝑉 → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵))))
28	27	com13 88	. . . 4 ⊢ (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))))
29	7, 28	syl 17	. . 3 ⊢ ((𝐵 ∈ Fin ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))))
30	29	ancoms 458	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))))
31		hashinf 14374	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
32	31	expcom 413	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin → (𝐴 ∈ 𝑉 → (♯‘𝐴) = +∞))
33	32	adantr 480	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ∈ 𝑉 → (♯‘𝐴) = +∞))
34	33	imp 406	. . . . . . 7 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) → (♯‘𝐴) = +∞)
35	34	adantr 480	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (♯‘𝐴) = +∞)
36		simpr 484	. . . . . . . 8 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
37		f1ofo 6855	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
38		focdmex 7980	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝑓:𝐴–onto→𝐵 → 𝐵 ∈ V))
39	38	imp 406	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ V)
40	36, 37, 39	syl2an 596	. . . . . . 7 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐵 ∈ V)
41		hashinf 14374	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
42	41	expcom 413	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ Fin → (𝐵 ∈ V → (♯‘𝐵) = +∞))
43	42	ad3antlr 731	. . . . . . 7 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (𝐵 ∈ V → (♯‘𝐵) = +∞))
44	40, 43	mpd 15	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (♯‘𝐵) = +∞)
45	35, 44	eqtr4d 2780	. . . . 5 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (♯‘𝐴) = (♯‘𝐵))
46	45	ex 412	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) → (𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))
47	46	exlimdv 1933	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))
48	47	ex 412	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))))
49	3, 6, 30, 48	4cases 1041	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3480  ^◡ccnv 5684  Rel wrel 5690  ⟶wf 6557  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  Fincfn 8985  +∞cpnf 11292  ♯chash 14369

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-hash 14370

This theorem is referenced by:  hashf1rn  14391  hasheqf1od  14392  2lgslem1  27438  nbedgusgr  29389  rusgrnumwrdl2  29604  wwlksnexthasheq  29923  rusgrnumwlkg  29997  numclwwlkqhash  30394  bj-finsumval0  37286  aks6d1c2  42131