Theorem dmct 10526

Description: The domain of a countable set is countable. The proof uses fodomnum 10060 rather than fodomg 10524, and so does not require ax-ac 10461. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Revised by Vincent Gonzalez, 24-Aug-2026.)

Assertion

Ref	Expression
dmct	⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω)

Proof of Theorem dmct

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmresv 6194	. 2 ⊢ dom (𝐴 ↾ V) = dom 𝐴
2		omelon 9625	. . . . . . 7 ⊢ ω ∈ On
3	2	a1i 11	. . . . . 6 ⊢ (𝐴 ≼ ω → ω ∈ On)
4		id 23	. . . . . 6 ⊢ (𝐴 ≼ ω → 𝐴 ≼ ω)
5		ondomen 10040	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐴 ≼ ω) → 𝐴 ∈ dom card)
6	3, 4, 5	syl2anc 596	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ dom card)
7		resss 5994	. . . . . 6 ⊢ (𝐴 ↾ V) ⊆ 𝐴
8	7	a1i 11	. . . . 5 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ⊆ 𝐴)
9		ssnum 10042	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ (𝐴 ↾ V) ⊆ 𝐴) → (𝐴 ↾ V) ∈ dom card)
10	6, 8, 9	syl2anc 596	. . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ∈ dom card)
11		fvex 6891	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
12		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))
13	11, 12	fnmpti 6675	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V)
14		dffn4 6795	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
15	13, 14	mpbi 233	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))
16		relres 5998	. . . . . 6 ⊢ Rel (𝐴 ↾ V)
17		reldm 8041	. . . . . 6 ⊢ (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
18		foeq3 6787	. . . . . 6 ⊢ (dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) → ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))))
19	16, 17, 18	mp2b 10	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
20	15, 19	mpbir 234	. . . 4 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V)
21		fodomnum 10060	. . . 4 ⊢ ((𝐴 ↾ V) ∈ dom card → ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V)))
22	10, 20, 21	mpisyl 22	. . 3 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V))
23		ctex 8969	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
24		ssdomg 9006	. . . . 5 ⊢ (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
25	23, 7, 24	mpisyl 22	. . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
26		domtr 9013	. . . 4 ⊢ (((𝐴 ↾ V) ≼ 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
27	25, 26	mpancom 701	. . 3 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
28		domtr 9013	. . 3 ⊢ ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω)
29	22, 27, 28	syl2anc 596	. 2 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω)
30	1, 29	eqbrtrrid 5141	1 ⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5655  ran crn 5656   ↾ cres 5657  Rel wrel 5660  Oncon0 6357   Fn wfn 6528  –onto→wfo 6531  ‘cfv 6533  ωcom 7862  1^st c1st 7984   ≼ cdom 8950  cardccrd 9940

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-card 9944  df-acn 9947

This theorem is used by:  rnct  10528