Theorem dmct 10641

Description: The domain of a countable set is countable. The proof uses fodomnum 10175 rather than fodomg 10639, and so does not require ax-ac 10576. (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 6213	. 2 ⊢ dom (𝐴 ↾ V) = dom 𝐴
2		omelon 9684	. . . . . . 7 ⊢ ω ∈ On
3	2	a1i 11	. . . . . 6 ⊢ (𝐴 ≼ ω → ω ∈ On)
4		id 23	. . . . . 6 ⊢ (𝐴 ≼ ω → 𝐴 ≼ ω)
5		ondomen 10155	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐴 ≼ ω) → 𝐴 ∈ dom card)
6	3, 4, 5	syl2anc 596	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ dom card)
7		resss 6008	. . . . . 6 ⊢ (𝐴 ↾ V) ⊆ 𝐴
8	7	a1i 11	. . . . 5 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ⊆ 𝐴)
9		ssnum 10157	. . . . 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 6929	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
12		eqid 2763	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))
13	11, 12	fnmpti 6712	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V)
14		dffn4 6832	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)⟶ontoran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
15	13, 14	mpbi 233	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)⟶ontoran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))
16		relres 6012	. . . . . 6 ⊢ Rel (𝐴 ↾ V)
17		reldm 8089	. . . . . 6 ⊢ (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
18		foeq3 6824	. . . . . 6 ⊢ (dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) → ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)⟶ontodom (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)⟶ontoran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))))
19	16, 17, 18	mp2b 10	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)⟶ontodom (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)⟶ontoran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
20	15, 19	mpbir 234	. . . 4 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)⟶ontodom (𝐴 ↾ V)
21		fodomnum 10175	. . . 4 ⊢ ((𝐴 ↾ V) ∈ dom card → ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)⟶ontodom (𝐴 ↾ V) → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V)))
22	10, 20, 21	mpisyl 22	. . 3 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V))
23		ctex 9028	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
24		ssdomg 9065	. . . . 5 ⊢ (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
25	23, 7, 24	mpisyl 22	. . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
26		domtr 9072	. . . 4 ⊢ (((𝐴 ↾ V) ≼ 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
27	25, 26	mpancom 701	. . 3 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
28		domtr 9072	. . 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 5147	1 ⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   class class class wbr 5108   ↦ cmpt 5192  dom cdm 5658  ran crn 5659   ↾ cres 5660  Rel wrel 5663  Oncon0 6388   Fn wfn 6561  ⟶ontowfo 6564  ‘cfv 6566  ωcom 7911  1^st c1st 8033   ≼ cdom 9009  cardccrd 10053

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-card 10057  df-acn 10060

This theorem is used by:  rnct  10643