Theorem dmct 10437

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 6153	. 2 ⊢ dom (𝐴 ↾ V) = dom 𝐴
2		resss 5956	. . . . 5 ⊢ (𝐴 ↾ V) ⊆ 𝐴
3		ctex 8896	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
4		ssexg 5265	. . . . 5 ⊢ (((𝐴 ↾ V) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝐴 ↾ V) ∈ V)
5	2, 3, 4	sylancr 587	. . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V)
6		fvex 6839	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
7		eqid 2729	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))
8	6, 7	fnmpti 6629	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V)
9		dffn4 6746	. . . . . 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 230	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))
11		relres 5960	. . . . . 6 ⊢ Rel (𝐴 ↾ V)
12		reldm 7986	. . . . . 6 ⊢ (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
13		foeq3 6738	. . . . . 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 231	. . . 4 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V)
16		fodomg 10435	. . . 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 8932	. . . . 5 ⊢ (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
19	3, 2, 18	mpisyl 21	. . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
20		domtr 8939	. . . 4 ⊢ (((𝐴 ↾ V) ≼ 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
21	19, 20	mpancom 688	. . 3 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
22		domtr 8939	. . 3 ⊢ ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω)
23	17, 21, 22	syl2anc 584	. 2 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω)
24	1, 23	eqbrtrrid 5131	1 ⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ⊆ wss 3905   class class class wbr 5095   ↦ cmpt 5176  dom cdm 5623  ran crn 5624   ↾ cres 5625  Rel wrel 5628   Fn wfn 6481  –onto→wfo 6484  ‘cfv 6486  ωcom 7806  1^st c1st 7929   ≼ cdom 8877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-ac2 10376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-card 9854  df-acn 9857  df-ac 10029

This theorem is referenced by:  rnct  10438