Theorem phplem4on 6867

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 6747	. . . . 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 5461	. . . . . . . 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 4595	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
7		nnon 4610	. . . . . . . . 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 4416	. . . . . . . 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 6793	. . . . . . 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 1239	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) ≈ 𝐴)
15	14	ensymd 6783	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓 “ 𝐴))
16		eloni 4376	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
17		orddif 4547	. . . . . . . . 9 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
18	16, 17	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 = (suc 𝐴 ∖ {𝐴}))
19	18	imaeq2d 4971	. . . . . . 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 5463	. . . . . . . . . 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 4417	. . . . . . . . . 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 5576	. . . . . . . . 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 3254	. . . . . . 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 4981	. . . . . . . . . . 11 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
29	28	eqcomi 2181	. . . . . . . . . 10 ⊢ ran 𝑓 = (𝑓 “ dom 𝑓)
30		f1ofo 5469	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵)
31		forn 5442	. . . . . . . . . . 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 5466	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
34	33	imaeq2d 4971	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
35	29, 32, 34	3eqtr3a 2234	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
36	35	difeq1d 3253	. . . . . . . 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 5468	. . . . . . . . . 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 5297	. . . . . . . . 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 2221	. . . . . 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 2210	. . . . 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 4030	. . . 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 5649	. . . . . . . 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 2247	. . . . . . . . 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 6856	. . . . . 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 6783	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ suc 𝐴 ≈ suc 𝐵) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵)
56		entr 6784	. . . 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 1898	. 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 1353  ∃wex 1492   ∈ wcel 2148   ∖ cdif 3127   ⊆ wss 3130  {csn 3593   class class class wbr 4004  Ord word 4363  Oncon0 4364  suc csuc 4366  ωcom 4590  ^◡ccnv 4626  dom cdm 4627  ran crn 4628   “ cima 4630  Fun wfun 5211   Fn wfn 5212  –1-1→wf1 5214  –onto→wfo 5215  –1-1-onto→wf1o 5216  ‘cfv 5217   ≈ cen 6738

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-er 6535  df-en 6741

This theorem is referenced by: (None)