Theorem axccdom 44899

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 483	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin)
2		simpr 483	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ Fin) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
3		axccdom.2	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅)
4	3	adantlr 713	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ Fin) ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅)
5	1, 2, 4	choicefi 44877	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ Fin) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
6		axccdom.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ≼ ω)
7	6	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → 𝑋 ≼ ω)
8		isfinite2 9342	. . . . . . 7 ⊢ (𝑋 ≺ ω → 𝑋 ∈ Fin)
9	8	con3i 154	. . . . . 6 ⊢ (¬ 𝑋 ∈ Fin → ¬ 𝑋 ≺ ω)
10	9	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → ¬ 𝑋 ≺ ω)
11	7, 10	jca 510	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → (𝑋 ≼ ω ∧ ¬ 𝑋 ≺ ω))
12		bren2 9022	. . . 4 ⊢ (𝑋 ≈ ω ↔ (𝑋 ≼ ω ∧ ¬ 𝑋 ≺ ω))
13	11, 12	sylibr 233	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → 𝑋 ≈ ω)
14		ctex 9002	. . . . . . 7 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V)
15	6, 14	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
16	15	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → 𝑋 ∈ V)
17		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → 𝑋 ≈ ω)
18		breq1 5159	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ≈ ω ↔ 𝑋 ≈ ω))
19		raleq 3316	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)))
20	19	exbidv 1917	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∃𝑔∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) ↔ ∃𝑔∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)))
21	18, 20	imbi12d 343	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ↔ (𝑋 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))))
22		ax-cc 10485	. . . . . 6 ⊢ (𝑥 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
23	21, 22	vtoclg 3550	. . . . 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 7246	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) ∈ V)
26	25	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) ∈ V)
27		fvex 6918	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑧) ∈ V
28	27	rgenw 3058	. . . . . . . . . . 11 ⊢ ∀𝑧 ∈ 𝑋 (𝑔‘𝑧) ∈ V
29		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))
30	29	fnmpt 6705	. . . . . . . . . . 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 1910	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝜑
34		nfra1 3276	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)
35	33, 34	nfan 1895	. . . . . . . . . 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 7025	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑔‘𝑧) ∈ V) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) = (𝑔‘𝑧))
39	36, 37, 38	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) = (𝑔‘𝑧))
40	39	adantl 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) = (𝑔‘𝑧))
41		rspa 3240	. . . . . . . . . . . . . 14 ⊢ ((∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) ∧ 𝑧 ∈ 𝑋) → (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
42	41	adantll 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
43	3	adantlr 713	. . . . . . . . . . . . 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 2829	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧)
47	46	ex 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → (𝑧 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
48	35, 47	ralrimi 3249	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧)
49	32, 48	jca 510	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
50		fneq1 6655	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → (𝑓 Fn 𝑋 ↔ (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋))
51		nfcv 2895	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧𝑓
52		nfmpt1 5264	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))
53	51, 52	nfeq 2909	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))
54		fveq1 6904	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → (𝑓‘𝑧) = ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧))
55	54	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
56	53, 55	ralbid 3265	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → (∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧 ↔ ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
57	50, 56	anbi12d 630	. . . . . . . . 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 3595	. . . . . . . 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 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≈ ω) ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
61	60	ex 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → (∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧)))
62	61	exlimdv 1929	. . . 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 589	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
65	5, 64	pm2.61dan 811	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2100   ≠ wne 2933  ∀wral 3054  Vcvv 3472  ∅c0 4335   class class class wbr 5156   ↦ cmpt 5239   Fn wfn 6553  ‘cfv 6558  ωcom 7884   ≈ cen 8979   ≼ cdom 8980   ≺ csdm 8981  Fincfn 8982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5293  ax-sep 5307  ax-nul 5314  ax-pow 5373  ax-pr 5437  ax-un 7751  ax-cc 10485

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-pss 3979  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-op 4643  df-uni 4919  df-int 4960  df-iun 5008  df-br 5157  df-opab 5219  df-mpt 5240  df-tr 5274  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5641  df-we 5643  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-pred 6317  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7433  df-om 7885  df-1st 8011  df-2nd 8012  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8742  df-en 8983  df-dom 8984  df-sdom 8985  df-fin 8986

This theorem is referenced by:  subsaliuncl  46049