Theorem phplem4on 6833

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 6713	. . . . 5 ⊢ (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
2	1	biimpi 119	. . . 4 ⊢ (suc 𝐴 ≈ suc 𝐵 → ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
3	2	adantl 275	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) → ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
4		f1of1 5431	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–1-1→suc 𝐵)
5	4	adantl 275	. . . . . . 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 4572	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
7		nnon 4587	. . . . . . . . 9 ⊢ (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
8	6, 7	syl 14	. . . . . . . 8 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
9	8	ad3antlr 485	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → suc 𝐵 ∈ On)
10		sssucid 4393	. . . . . . . 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 523	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ∈ On)
13		f1imaen2g 6759	. . . . . . 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 1229	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) ≈ 𝐴)
15	14	ensymd 6749	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓 “ 𝐴))
16		eloni 4353	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
17		orddif 4524	. . . . . . . . 9 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
18	16, 17	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 = (suc 𝐴 ∖ {𝐴}))
19	18	imaeq2d 4946	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
20	19	ad3antrrr 484	. . . . . 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 5433	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓 Fn suc 𝐴)
22	21	adantl 275	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝑓 Fn suc 𝐴)
23		sucidg 4394	. . . . . . . . . 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 5545	. . . . . . . . 9 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
26	22, 24, 25	syl2anc 409	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
27	26	difeq2d 3240	. . . . . . 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 4956	. . . . . . . . . . 11 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
29	28	eqcomi 2169	. . . . . . . . . 10 ⊢ ran 𝑓 = (𝑓 “ dom 𝑓)
30		f1ofo 5439	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵)
31		forn 5413	. . . . . . . . . . 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 5436	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
34	33	imaeq2d 4946	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
35	29, 32, 34	3eqtr3a 2223	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
36	35	difeq1d 3239	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}))
37	36	adantl 275	. . . . . . 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 5438	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴–onto→suc 𝐵 ∧ Fun ◡𝑓))
39	38	simprbi 273	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → Fun ◡𝑓)
40		imadif 5268	. . . . . . . . 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 275	. . . . . . 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 2209	. . . . . 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 2198	. . . . 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 4008	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
46		simpllr 524	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐵 ∈ ω)
47		fnfvelrn 5617	. . . . . . . 8 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → (𝑓‘𝐴) ∈ ran 𝑓)
48	22, 24, 47	syl2anc 409	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓‘𝐴) ∈ ran 𝑓)
49	31	eleq2d 2236	. . . . . . . . 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 275	. . . . . . 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 146	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓‘𝐴) ∈ suc 𝐵)
53		phplem3g 6822	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ (𝑓‘𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
54	46, 52, 53	syl2anc 409	. . . . 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 6749	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵)
56		entr 6750	. . . 4 ⊢ ((𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
57	45, 55, 56	syl2anc 409	. . 3 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ 𝐵)
58	3, 57	exlimddv 1886	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) → 𝐴 ≈ 𝐵)
59	58	ex 114	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136   ∖ cdif 3113   ⊆ wss 3116  {csn 3576   class class class wbr 3982  Ord word 4340  Oncon0 4341  suc csuc 4343  ωcom 4567  ^◡ccnv 4603  dom cdm 4604  ran crn 4605   “ cima 4607  Fun wfun 5182   Fn wfn 5183  –1-1→wf1 5185  –onto→wfo 5186  –1-1-onto→wf1o 5187  ‘cfv 5188   ≈ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-er 6501  df-en 6707

This theorem is referenced by: (None)