Theorem phplem4on 7049

Description: Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)

Assertion

Ref	Expression
phplem4on	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

Proof of Theorem phplem4on

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6912	. . . . 5 ⊢ (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
2	1	biimpi 120	. . . 4 ⊢ (suc 𝐴 ≈ suc 𝐵 → ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
3	2	adantl 277	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) → ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
4		f1of1 5579	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–1-1→suc 𝐵)
5	4	adantl 277	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝑓:suc 𝐴–1-1→suc 𝐵)
6		peano2 4691	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
7		nnon 4706	. . . . . . . . 9 ⊢ (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
8	6, 7	syl 14	. . . . . . . 8 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
9	8	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → suc 𝐵 ∈ On)
10		sssucid 4510	. . . . . . . 8 ⊢ 𝐴 ⊆ suc 𝐴
11	10	a1i 9	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ⊆ suc 𝐴)
12		simplll 533	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ∈ On)
13		f1imaen2g 6962	. . . . . . 7 ⊢ (((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ suc 𝐵 ∈ On) ∧ (𝐴 ⊆ suc 𝐴 ∧ 𝐴 ∈ On)) → (𝑓 “ 𝐴) ≈ 𝐴)
14	5, 9, 11, 12, 13	syl22anc 1272	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) ≈ 𝐴)
15	14	ensymd 6952	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓 “ 𝐴))
16		eloni 4470	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
17		orddif 4643	. . . . . . . . 9 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
18	16, 17	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 = (suc 𝐴 ∖ {𝐴}))
19	18	imaeq2d 5074	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
20	19	ad3antrrr 492	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
21		f1ofn 5581	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓 Fn suc 𝐴)
22	21	adantl 277	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝑓 Fn suc 𝐴)
23		sucidg 4511	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
24	12, 23	syl 14	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ∈ suc 𝐴)
25		fnsnfv 5701	. . . . . . . . 9 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
26	22, 24, 25	syl2anc 411	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
27	26	difeq2d 3323	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
28		imadmrn 5084	. . . . . . . . . . 11 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
29	28	eqcomi 2233	. . . . . . . . . 10 ⊢ ran 𝑓 = (𝑓 “ dom 𝑓)
30		f1ofo 5587	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵)
31		forn 5559	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
32	30, 31	syl 14	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
33		f1odm 5584	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
34	33	imaeq2d 5074	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
35	29, 32, 34	3eqtr3a 2286	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
36	35	difeq1d 3322	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}))
37	36	adantl 277	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}))
38		dff1o3 5586	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴–onto→suc 𝐵 ∧ Fun ◡𝑓))
39	38	simprbi 275	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → Fun ◡𝑓)
40		imadif 5407	. . . . . . . . 9 ⊢ (Fun ◡𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
41	39, 40	syl 14	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
42	41	adantl 277	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
43	27, 37, 42	3eqtr4rd 2273	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
44	20, 43	eqtrd 2262	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
45	15, 44	breqtrd 4112	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
46		simpllr 534	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐵 ∈ ω)
47		fnfvelrn 5775	. . . . . . . 8 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → (𝑓‘𝐴) ∈ ran 𝑓)
48	22, 24, 47	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓‘𝐴) ∈ ran 𝑓)
49	31	eleq2d 2299	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
50	30, 49	syl 14	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
51	50	adantl 277	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
52	48, 51	mpbid 147	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓‘𝐴) ∈ suc 𝐵)
53		phplem3g 7037	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ (𝑓‘𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
54	46, 52, 53	syl2anc 411	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
55	54	ensymd 6952	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵)
56		entr 6953	. . . 4 ⊢ ((𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
57	45, 55, 56	syl2anc 411	. . 3 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ 𝐵)
58	3, 57	exlimddv 1945	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) → 𝐴 ≈ 𝐵)
59	58	ex 115	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200   ∖ cdif 3195   ⊆ wss 3198  {csn 3667   class class class wbr 4086  Ord word 4457  Oncon0 4458  suc csuc 4460  ωcom 4686  ^◡ccnv 4722  dom cdm 4723  ran crn 4724   “ cima 4726  Fun wfun 5318   Fn wfn 5319  –1-1→wf1 5321  –onto→wfo 5322  –1-1-onto→wf1o 5323  ‘cfv 5324   ≈ cen 6902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-er 6697  df-en 6905

This theorem is referenced by: (None)