Theorem hasheqf1oi 13562

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 13559	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
2	1	biimprd 249	. . 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 13560	. . . 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 13560	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ ¬ 𝐴 ∈ Fin) → ¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)
8		19.41v 1927	. . . . . . 7 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉))
9		f1orel 6486	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → Rel 𝑓)
10	9	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → Rel 𝑓)
11		f1ocnvb 6496	. . . . . . . . . . . 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 6483	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
14		fex 6855	. . . . . . . . . . . . . 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 7485	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ V → ◡𝑓 ∈ V)
17		f1oeq1 6472	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵–1-1-onto→𝐴 ↔ ◡𝑓:𝐵–1-1-onto→𝐴))
18	17	spcegv 3540	. . . . . . . . . . . . 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 241	. . . . . . . . . 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 665	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵)))
25	24	exlimiv 1908	. . . . . . 7 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵)))
26	8, 25	sylbir 236	. . . . . 6 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (♯‘𝐴) = (♯‘𝐵)))
27	26	ex 413	. . . . 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 459	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))))
31		hashinf 13545	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
32	31	expcom 414	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin → (𝐴 ∈ 𝑉 → (♯‘𝐴) = +∞))
33	32	adantr 481	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ∈ 𝑉 → (♯‘𝐴) = +∞))
34	33	imp 407	. . . . . . 7 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) → (♯‘𝐴) = +∞)
35	34	adantr 481	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (♯‘𝐴) = +∞)
36		simpr 485	. . . . . . . 8 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
37		f1ofo 6490	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
38		focdmex 13561	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ V)
39	36, 37, 38	syl2an 595	. . . . . . 7 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐵 ∈ V)
40		hashinf 13545	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
41	40	expcom 414	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ Fin → (𝐵 ∈ V → (♯‘𝐵) = +∞))
42	41	ad3antlr 727	. . . . . . 7 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (𝐵 ∈ V → (♯‘𝐵) = +∞))
43	39, 42	mpd 15	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (♯‘𝐵) = +∞)
44	35, 43	eqtr4d 2834	. . . . 5 ⊢ ((((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (♯‘𝐴) = (♯‘𝐵))
45	44	ex 413	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) → (𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))
46	45	exlimdv 1911	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ 𝐴 ∈ 𝑉) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))
47	46	ex 413	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))))
48	3, 6, 30, 47	4cases 1033	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081  Vcvv 3437  ^◡ccnv 5442  Rel wrel 5448  ⟶wf 6221  –onto→wfo 6223  –1-1-onto→wf1o 6224  ‘cfv 6225  Fincfn 8357  +∞cpnf 10518  ♯chash 13540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-n0 11746  df-z 11830  df-uz 12094  df-hash 13541

This theorem is referenced by:  hashf1rn  13563  hasheqf1od  13564  2lgslem1  25652  nbedgusgr  26837  rusgrnumwrdl2  27051  wwlksnexthasheq  27368  rusgrnumwlkg  27443  numclwwlkqhash  27846  bj-finsumval0  34119