Theorem hasheqf1oi 14323

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 14321	. . . 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 14322	. . . 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 14322	. . . 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 6806	. . . . . . . . . . . . 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 6816	. . . . . . . . . . . 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 6803	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
14		fex 7203	. . . . . . . . . . . . . 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 7903	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ V → ◡𝑓 ∈ V)
17		f1oeq1 6791	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵–1-1-onto→𝐴 ↔ ◡𝑓:𝐵–1-1-onto→𝐴))
18	17	spcegv 3566	. . . . . . . . . . . . 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 14307	. . . . . . . . . 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 6810	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
38		focdmex 7937	. . . . . . . . 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 14307	. . . . . . . . 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 2768	. . . . 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 1040	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 2109  Vcvv 3450  ^◡ccnv 5640  Rel wrel 5646  ⟶wf 6510  –onto→wfo 6512  –1-1-onto→wf1o 6513  ‘cfv 6514  Fincfn 8921  +∞cpnf 11212  ♯chash 14302

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-hash 14303

This theorem is referenced by:  hashf1rn  14324  hasheqf1od  14325  2lgslem1  27312  nbedgusgr  29306  rusgrnumwrdl2  29521  wwlksnexthasheq  29840  rusgrnumwlkg  29914  numclwwlkqhash  30311  bj-finsumval0  37280  aks6d1c2  42125