Theorem fnct 10556

Description: If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)

Assertion

Ref	Expression
fnct	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω)

Proof of Theorem fnct

Step	Hyp	Ref	Expression
1		ctex 8983	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
2	1	adantl 481	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐴 ∈ V)
3		fndm 6646	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	eleq1d 2820	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
5	4	adantr 480	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
6	2, 5	mpbird 257	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → dom 𝐹 ∈ V)
7		fnfun 6643	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
8	7	adantr 480	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → Fun 𝐹)
9		funrnex 7957	. . . . 5 ⊢ (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
10	6, 8, 9	sylc 65	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ∈ V)
11	2, 10	xpexd 7750	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V)
12		simpl 482	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 Fn 𝐴)
13		dffn3 6723	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
14	12, 13	sylib 218	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹)
15		fssxp 6738	. . . 4 ⊢ (𝐹:𝐴⟶ran 𝐹 → 𝐹 ⊆ (𝐴 × ran 𝐹))
16	14, 15	syl 17	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ⊆ (𝐴 × ran 𝐹))
17		ssdomg 9019	. . 3 ⊢ ((𝐴 × ran 𝐹) ∈ V → (𝐹 ⊆ (𝐴 × ran 𝐹) → 𝐹 ≼ (𝐴 × ran 𝐹)))
18	11, 16, 17	sylc 65	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ (𝐴 × ran 𝐹))
19		xpdom1g 9088	. . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
20	10, 19	sylancom 588	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
21		omex 9662	. . . . 5 ⊢ ω ∈ V
22		fnrndomg 10555	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴))
23	2, 12, 22	sylc 65	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ 𝐴)
24		domtr 9026	. . . . . 6 ⊢ ((ran 𝐹 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω)
25	23, 24	sylancom 588	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω)
26		xpdom2g 9087	. . . . 5 ⊢ ((ω ∈ V ∧ ran 𝐹 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
27	21, 25, 26	sylancr 587	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
28		domtr 9026	. . . 4 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ran 𝐹) ∧ (ω × ran 𝐹) ≼ (ω × ω)) → (𝐴 × ran 𝐹) ≼ (ω × ω))
29	20, 27, 28	syl2anc 584	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ω))
30		xpomen 10034	. . 3 ⊢ (ω × ω) ≈ ω
31		domentr 9032	. . 3 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ran 𝐹) ≼ ω)
32	29, 30, 31	sylancl 586	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ ω)
33		domtr 9026	. 2 ⊢ ((𝐹 ≼ (𝐴 × ran 𝐹) ∧ (𝐴 × ran 𝐹) ≼ ω) → 𝐹 ≼ ω)
34	18, 32, 33	syl2anc 584	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124   × cxp 5657  dom cdm 5659  ran crn 5660  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ωcom 7866   ≈ cen 8961   ≼ cdom 8962

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-ac2 10482

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9529  df-card 9958  df-acn 9961  df-ac 10135

This theorem is referenced by:  mptct  10557  mpocti  32698  mptctf  32700  omssubadd  34337