Theorem dmct 10474

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 6181	. 2 ⊢ dom (𝐴 ↾ V) = dom 𝐴
2		resss 5983	. . . . 5 ⊢ (𝐴 ↾ V) ⊆ 𝐴
3		ctex 8937	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
4		ssexg 5276	. . . . 5 ⊢ (((𝐴 ↾ V) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝐴 ↾ V) ∈ V)
5	2, 3, 4	sylancr 596	. . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V)
6		fvex 6874	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
7		eqid 2761	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))
8	6, 7	fnmpti 6658	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V)
9		dffn4 6778	. . . . . 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 5987	. . . . . 6 ⊢ Rel (𝐴 ↾ V)
12		reldm 8019	. . . . . 6 ⊢ (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))
13		foeq3 6770	. . . . . 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 10472	. . . 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 8974	. . . . 5 ⊢ (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
19	3, 2, 18	mpisyl 21	. . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
20		domtr 8981	. . . 4 ⊢ (((𝐴 ↾ V) ≼ 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
21	19, 20	mpancom 698	. . 3 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
22		domtr 8981	. . 3 ⊢ ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω)
23	17, 21, 22	syl2anc 593	. 2 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω)
24	1, 23	eqbrtrrid 5133	1 ⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3902   class class class wbr 5097   ↦ cmpt 5178  dom cdm 5643  ran crn 5644   ↾ cres 5645  Rel wrel 5648   Fn wfn 6510  –onto→wfo 6513  ‘cfv 6515  ωcom 7840  1^st c1st 7962   ≼ cdom 8918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-ac2 10413

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-isom 6524  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-er 8671  df-map 8803  df-en 8921  df-dom 8922  df-card 9890  df-acn 9893  df-ac 10065

This theorem is referenced by:  rnct  10475