Theorem axccdom 45462

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 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin)
2		simpr 484	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ Fin) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
3		axccdom.2	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅)
4	3	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ Fin) ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅)
5	1, 2, 4	choicefi 45440	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ Fin) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
6		axccdom.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ≼ ω)
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → 𝑋 ≼ ω)
8		isfinite2 9198	. . . . . . 7 ⊢ (𝑋 ≺ ω → 𝑋 ∈ Fin)
9	8	con3i 154	. . . . . 6 ⊢ (¬ 𝑋 ∈ Fin → ¬ 𝑋 ≺ ω)
10	9	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → ¬ 𝑋 ≺ ω)
11	7, 10	jca 511	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → (𝑋 ≼ ω ∧ ¬ 𝑋 ≺ ω))
12		bren2 8920	. . . 4 ⊢ (𝑋 ≈ ω ↔ (𝑋 ≼ ω ∧ ¬ 𝑋 ≺ ω))
13	11, 12	sylibr 234	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ Fin) → 𝑋 ≈ ω)
14		ctex 8900	. . . . . . 7 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V)
15	6, 14	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
16	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → 𝑋 ∈ V)
17		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≈ ω) → 𝑋 ≈ ω)
18		breq1 5101	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ≈ ω ↔ 𝑋 ≈ ω))
19		raleq 3293	. . . . . . . 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 344	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ↔ (𝑋 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))))
22		ax-cc 10345	. . . . . 6 ⊢ (𝑥 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
23	21, 22	vtoclg 3511	. . . . 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 7170	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) ∈ V)
26	25	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) ∈ V)
27		fvex 6847	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑧) ∈ V
28	27	rgenw 3055	. . . . . . . . . . 11 ⊢ ∀𝑧 ∈ 𝑋 (𝑔‘𝑧) ∈ V
29		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))
30	29	fnmpt 6632	. . . . . . . . . . 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 3260	. . . . . . . . . . 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 6952	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑔‘𝑧) ∈ V) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) = (𝑔‘𝑧))
39	36, 37, 38	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) = (𝑔‘𝑧))
40	39	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) = (𝑔‘𝑧))
41		rspa 3225	. . . . . . . . . . . . . 14 ⊢ ((∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧) ∧ 𝑧 ∈ 𝑋) → (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
42	41	adantll 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
43	3	adantlr 715	. . . . . . . . . . . . 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 2836	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) ∧ 𝑧 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧)
47	46	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → (𝑧 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
48	35, 47	ralrimi 3234	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧)
49	32, 48	jca 511	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
50		fneq1 6583	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → (𝑓 Fn 𝑋 ↔ (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) Fn 𝑋))
51		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧𝑓
52		nfmpt1 5197	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))
53	51, 52	nfeq 2912	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))
54		fveq1 6833	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → (𝑓‘𝑧) = ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧))
55	54	eleq1d 2821	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
56	53, 55	ralbid 3249	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧)) → (∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧 ↔ ∀𝑧 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (𝑔‘𝑧))‘𝑧) ∈ 𝑧))
57	50, 56	anbi12d 632	. . . . . . . . 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 3551	. . . . . . . 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 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≈ ω) ∧ ∀𝑧 ∈ 𝑋 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
61	60	ex 412	. . . . 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 591	. 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 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  Vcvv 3440  ∅c0 4285   class class class wbr 5098   ↦ cmpt 5179   Fn wfn 6487  ‘cfv 6492  ωcom 7808   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cc 10345

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887

This theorem is referenced by:  subsaliuncl  46598