Theorem fnct 10491

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 8940	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
2	1	adantl 485	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐴 ∈ V)
3		fndm 6620	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	eleq1d 2846	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
5	4	adantr 484	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
6	2, 5	mpbird 259	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → dom 𝐹 ∈ V)
7		fnfun 6617	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
8	7	adantr 484	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → Fun 𝐹)
9		funrnex 7931	. . . . 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 7730	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V)
12		dffn3 6700	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
13	12	birani 507	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹)
14		fssxp 6715	. . . 4 ⊢ (𝐹:𝐴⟶ran 𝐹 → 𝐹 ⊆ (𝐴 × ran 𝐹))
15	13, 14	syl 17	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ⊆ (𝐴 × ran 𝐹))
16		ssdomg 8977	. . 3 ⊢ ((𝐴 × ran 𝐹) ∈ V → (𝐹 ⊆ (𝐴 × ran 𝐹) → 𝐹 ≼ (𝐴 × ran 𝐹)))
17	11, 15, 16	sylc 65	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ (𝐴 × ran 𝐹))
18		xpdom1g 9042	. . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
19	10, 18	sylancom 597	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
20		omex 9595	. . . . 5 ⊢ ω ∈ V
21		simpl 486	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 Fn 𝐴)
22		fnrndomg 10490	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴))
23	2, 21, 22	sylc 65	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ 𝐴)
24		domtr 8984	. . . . . 6 ⊢ ((ran 𝐹 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω)
25	23, 24	sylancom 597	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω)
26		xpdom2g 9041	. . . . 5 ⊢ ((ω ∈ V ∧ ran 𝐹 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
27	20, 25, 26	sylancr 596	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
28		domtr 8984	. . . 4 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ran 𝐹) ∧ (ω × ran 𝐹) ≼ (ω × ω)) → (𝐴 × ran 𝐹) ≼ (ω × ω))
29	19, 27, 28	syl2anc 593	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ω))
30		xpomen 9968	. . 3 ⊢ (ω × ω) ≈ ω
31		domentr 8990	. . 3 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ran 𝐹) ≼ ω)
32	29, 30, 31	sylancl 595	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ ω)
33		domtr 8984	. 2 ⊢ ((𝐹 ≼ (𝐴 × ran 𝐹) ∧ (𝐴 × ran 𝐹) ≼ ω) → 𝐹 ≼ ω)
34	17, 32, 33	syl2anc 593	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904   class class class wbr 5099   × cxp 5643  dom cdm 5645  ran crn 5646  Fun wfun 6511   Fn wfn 6512  ⟶wf 6513  ωcom 7842   ≈ cen 8920   ≼ cdom 8921

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-ac2 10417

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 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-oi 9455  df-card 9894  df-acn 9897  df-ac 10069

This theorem is referenced by:  mptct  10492  mpocti  32866  mptctf  32868  omssubadd  34558