Theorem hasheqf1oiOLD 13141

Description: Obsolete version of hasheqf1oi 13140 as of 4-May-2021. (Contributed by Alexander van der Vekens, 25-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
hasheqf1oiOLD	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (#‘𝐴) = (#‘𝐵)))

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

Proof of Theorem hasheqf1oiOLD

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hasheqf1o 13137	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) = (#‘𝐵) ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
2	1	biimprd 238	. . 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 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (#‘𝐴) = (#‘𝐵))))
4		fiinfnf1o 13138	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
5	4	pm2.21d 118	. . 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 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (#‘𝐴) = (#‘𝐵))))
7		fiinfnf1o 13138	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ ¬ 𝐴 ∈ Fin) → ¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)
8		19.41v 1914	. . . . . . 7 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)))
9		f1orel 6140	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → Rel 𝑓)
10	9	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → Rel 𝑓)
11		f1ocnvb 6150	. . . . . . . . . . . 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 ∧ 𝐵 ∈ 𝑊)) → (𝑓:𝐴–1-1-onto→𝐵 ↔ ◡𝑓:𝐵–1-1-onto→𝐴))
13		f1of 6137	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
14	13	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝑓:𝐴⟶𝐵)
15		simprl 794	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
16		simprr 796	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐵 ∈ 𝑊)
17		fex2 7121	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑓 ∈ V)
18	14, 15, 16, 17	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝑓 ∈ V)
19		cnvexg 7112	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ V → ◡𝑓 ∈ V)
20		f1oeq1 6127	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵–1-1-onto→𝐴 ↔ ◡𝑓:𝐵–1-1-onto→𝐴))
21	20	spcegv 3294	. . . . . . . . . . . . 13 ⊢ (◡𝑓 ∈ V → (◡𝑓:𝐵–1-1-onto→𝐴 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴))
22	18, 19, 21	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (◡𝑓:𝐵–1-1-onto→𝐴 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴))
23		pm2.24 121	. . . . . . . . . . . 12 ⊢ (∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (#‘𝐴) = (#‘𝐵)))
24	22, 23	syl6 35	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (◡𝑓:𝐵–1-1-onto→𝐴 → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (#‘𝐴) = (#‘𝐵))))
25	12, 24	sylbid 230	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝑓:𝐴–1-1-onto→𝐵 → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (#‘𝐴) = (#‘𝐵))))
26	25	com12 32	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (#‘𝐴) = (#‘𝐵))))
27	26	anabsi5 858	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (#‘𝐴) = (#‘𝐵)))
28	27	exlimiv 1858	. . . . . . 7 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (#‘𝐴) = (#‘𝐵)))
29	8, 28	sylbir 225	. . . . . 6 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (#‘𝐴) = (#‘𝐵)))
30	29	ex 450	. . . . 5 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → (#‘𝐴) = (#‘𝐵))))
31	30	com13 88	. . . 4 ⊢ (¬ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (#‘𝐴) = (#‘𝐵))))
32	7, 31	syl 17	. . 3 ⊢ ((𝐵 ∈ Fin ∧ ¬ 𝐴 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (#‘𝐴) = (#‘𝐵))))
33	32	ancoms 469	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (#‘𝐴) = (#‘𝐵))))
34		hashinf 13122	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞)
35	34	expcom 451	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin → (𝐴 ∈ 𝑉 → (#‘𝐴) = +∞))
36	35	adantr 481	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ∈ 𝑉 → (#‘𝐴) = +∞))
37	36	com12 32	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) = +∞))
38	37	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) = +∞))
39	38	impcom 446	. . . . 5 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (#‘𝐴) = +∞)
40		hashinf 13122	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑊 ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
41	40	expcom 451	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ Fin → (𝐵 ∈ 𝑊 → (#‘𝐵) = +∞))
42	41	adantl 482	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (𝐵 ∈ 𝑊 → (#‘𝐵) = +∞))
43	42	com12 32	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞))
44	43	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞))
45	44	impcom 446	. . . . 5 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (#‘𝐵) = +∞)
46	39, 45	eqtr4d 2659	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (#‘𝐴) = (#‘𝐵))
47	46	a1d 25	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (#‘𝐴) = (#‘𝐵)))
48	47	ex 450	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (#‘𝐴) = (#‘𝐵))))
49	3, 6, 33, 48	4cases 990	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (#‘𝐴) = (#‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200  ^◡ccnv 5113  Rel wrel 5119  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  Fincfn 7955  +∞cpnf 10071  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-hash 13118

This theorem is referenced by:  hashf1rnOLD  13143