Theorem axccdom 42221

Description: Relax the constraint on ax-cc to dominance instead of equinumerosity. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
axccdom.1	⊢ (𝜑 → 𝑋 ≼ ω)
axccdom.2	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅)

Assertion

Ref	Expression
axccdom	⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))

Distinct variable groups:   𝑓,𝑋,𝑧   𝜑,𝑧

Allowed substitution hint:   𝜑(𝑓)

Proof of Theorem axccdom

Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin)
2		simpr 488	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ Fin) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
3		axccdom.2	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅)
4	3	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ Fin) ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅)
5	1, 2, 4	choicefi 42199	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ Fin) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
6		axccdom.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ≼ ω)
7	6	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → 𝑋 ≼ ω)
8		isfinite2 8809	. . . . . . 7 ⊢ (𝑋 ≺ ω → 𝑋 ∈ Fin)
9	8	con3i 157	. . . . . 6 ⊢ (¬ 𝑋 ∈ Fin → ¬ 𝑋 ≺ ω)
10	9	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → ¬ 𝑋 ≺ ω)
11	7, 10	jca 515	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → (𝑋 ≼ ω ∧ ¬ 𝑋 ≺ ω))
12		bren2 8558	. . . 4 ⊢ (𝑋 ≈ ω ↔ (𝑋 ≼ ω ∧ ¬ 𝑋 ≺ ω))
13	11, 12	sylibr 237	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → 𝑋 ≈ ω)
14		ctex 8542	. . . . . . 7 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V)
15	6, 14	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
16	15	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → 𝑋 ∈ V)
17		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → 𝑋 ≈ ω)
18		breq1 5035	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ≈ ω ↔ 𝑋 ≈ ω))
19		raleq 3323	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)))
20	19	exbidv 1922	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∃𝑔∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) ↔ ∃𝑔∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)))
21	18, 20	imbi12d 348	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ↔ (𝑋 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))))
22		ax-cc 9895	. . . . . 6 ⊢ (𝑥 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
23	21, 22	vtoclg 3485	. . . . 5 ⊢ (𝑋 ∈ V → (𝑋 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)))
24	16, 17, 23	sylc 65	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → ∃𝑔∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
25	15	mptexd 6978	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) ∈ V)
26	25	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) ∈ V)
27		fvex 6671	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑧) ∈ V
28	27	rgenw 3082	. . . . . . . . . . 11 ⊢ ∀𝑧 ∈ 𝑋 (𝑔‘𝑧) ∈ V
29		eqid 2758	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))
30	29	fnmpt 6471	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑋 (𝑔‘𝑧) ∈ V → (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋)
31	28, 30	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋
32	31	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋)
33		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝜑
34		nfra1 3147	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)
35	33, 34	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
36		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → 𝑧 ∈ 𝑋)
37	27	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → (𝑔‘𝑧) ∈ V)
38	29	fvmpt2 6770	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑔‘𝑧) ∈ V) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) = (𝑔‘𝑧))
39	36, 37, 38	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) = (𝑔‘𝑧))
40	39	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) = (𝑔‘𝑧))
41		rspa 3135	. . . . . . . . . . . . . 14 ⊢ ((∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) ∧ 𝑧 ∈ 𝑋) → (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
42	41	adantll 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
43	3	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅)
44		id 22	. . . . . . . . . . . . 13 ⊢ ((𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) → (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
45	42, 43, 44	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → (𝑔‘𝑧) ∈ 𝑧)
46	40, 45	eqeltrd 2852	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧)
47	46	ex 416	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → (𝑧 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
48	35, 47	ralrimi 3144	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧)
49	32, 48	jca 515	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
50		fneq1 6425	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → (𝑓 Fn 𝑋 ↔ (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋))
51		nfcv 2919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧𝑓
52		nfmpt1 5130	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))
53	51, 52	nfeq 2932	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))
54		fveq1 6657	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → (𝑓‘𝑧) = ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧))
55	54	eleq1d 2836	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
56	53, 55	ralbid 3159	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → (∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧 ↔ ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
57	50, 56	anbi12d 633	. . . . . . . . 9 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → ((𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧) ↔ ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧)))
58	57	spcegv 3515	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) ∈ V → (((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧)))
59	26, 49, 58	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
60	59	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≈ ω) ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
61	60	ex 416	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → (∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧)))
62	61	exlimdv 1934	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → (∃𝑔∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧)))
63	24, 62	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
64	13, 63	syldan 594	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
65	5, 64	pm2.61dan 812	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  Vcvv 3409  ∅c0 4225   class class class wbr 5032   ↦ cmpt 5112   Fn wfn 6330  ‘cfv 6335  ωcom 7579   ≈ cen 8524   ≼ cdom 8525   ≺ csdm 8526  Fincfn 8527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cc 9895

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531

This theorem is referenced by:  subsaliuncl  43364