Theorem phplem4on 6845

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 6725	. . . . 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 5441	. . . . . . . 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 4579	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
7		nnon 4594	. . . . . . . . 9 ⊢ (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
8	6, 7	syl 14	. . . . . . . 8 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
9	8	ad3antlr 490	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → suc 𝐵 ∈ On)
10		sssucid 4400	. . . . . . . 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 528	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ∈ On)
13		f1imaen2g 6771	. . . . . . 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 1234	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) ≈ 𝐴)
15	14	ensymd 6761	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓 “ 𝐴))
16		eloni 4360	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
17		orddif 4531	. . . . . . . . 9 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
18	16, 17	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 = (suc 𝐴 ∖ {𝐴}))
19	18	imaeq2d 4953	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
20	19	ad3antrrr 489	. . . . . 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 5443	. . . . . . . . . 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 4401	. . . . . . . . . 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 5555	. . . . . . . . 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 3245	. . . . . . 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 4963	. . . . . . . . . . 11 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
29	28	eqcomi 2174	. . . . . . . . . 10 ⊢ ran 𝑓 = (𝑓 “ dom 𝑓)
30		f1ofo 5449	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵)
31		forn 5423	. . . . . . . . . . 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 5446	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
34	33	imaeq2d 4953	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
35	29, 32, 34	3eqtr3a 2227	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
36	35	difeq1d 3244	. . . . . . . 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 5448	. . . . . . . . . 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 5278	. . . . . . . . 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 2214	. . . . . 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 2203	. . . . 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 4015	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
46		simpllr 529	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐵 ∈ ω)
47		fnfvelrn 5628	. . . . . . . 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 2240	. . . . . . . . 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 6834	. . . . . 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 6761	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵)
56		entr 6762	. . . 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 1891	. 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 1348  ∃wex 1485   ∈ wcel 2141   ∖ cdif 3118   ⊆ wss 3121  {csn 3583   class class class wbr 3989  Ord word 4347  Oncon0 4348  suc csuc 4350  ωcom 4574  ^◡ccnv 4610  dom cdm 4611  ran crn 4612   “ cima 4614  Fun wfun 5192   Fn wfn 5193  –1-1→wf1 5195  –onto→wfo 5196  –1-1-onto→wf1o 5197  ‘cfv 5198   ≈ cen 6716

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719

This theorem is referenced by: (None)