Theorem dmct 9946

Description: The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)

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 6057	. 2 ⊢ dom (𝐴 ↾ V) = dom 𝐴
2		resss 5878	. . . . 5 ⊢ (𝐴 ↾ V) ⊆ 𝐴
3		ctex 8524	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
4		ssexg 5227	. . . . 5 ⊢ (((𝐴 ↾ V) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝐴 ↾ V) ∈ V)
5	2, 3, 4	sylancr 589	. . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V)
6		fvex 6683	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
7		eqid 2821	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))
8	6, 7	fnmpti 6491	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V)
9		dffn4 6596	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
10	8, 9	mpbi 232	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))
11		relres 5882	. . . . . 6 ⊢ Rel (𝐴 ↾ V)
12		reldm 7743	. . . . . 6 ⊢ (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
13		foeq3 6588	. . . . . 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))))
14	11, 12, 13	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)))
15	10, 14	mpbir 233	. . . 4 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V)
16		fodomg 9945	. . . 4 ⊢ ((𝐴 ↾ V) ∈ V → ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V)))
17	5, 15, 16	mpisyl 21	. . 3 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V))
18		ssdomg 8555	. . . . 5 ⊢ (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
19	3, 2, 18	mpisyl 21	. . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
20		domtr 8562	. . . 4 ⊢ (((𝐴 ↾ V) ≼ 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
21	19, 20	mpancom 686	. . 3 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
22		domtr 8562	. . 3 ⊢ ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω)
23	17, 21, 22	syl2anc 586	. 2 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω)
24	1, 23	eqbrtrrid 5102	1 ⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ⊆ wss 3936   class class class wbr 5066   ↦ cmpt 5146  dom cdm 5555  ran crn 5556   ↾ cres 5557  Rel wrel 5560   Fn wfn 6350  –onto→wfo 6353  ‘cfv 6355  ωcom 7580  1^st c1st 7687   ≼ cdom 8507

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-ac2 9885

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-card 9368  df-acn 9371  df-ac 9542

This theorem is referenced by:  rnct  9947