Theorem fnct 10554

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 8977	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
2	1	adantl 481	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐴 ∈ V)
3		fndm 6651	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	eleq1d 2814	. . . . . . 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 6648	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
8	7	adantr 480	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → Fun 𝐹)
9		funrnex 7951	. . . . 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 7747	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V)
12		simpl 482	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 Fn 𝐴)
13		dffn3 6729	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
14	12, 13	sylib 217	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹)
15		fssxp 6745	. . . 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 9014	. . 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 9087	. . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
20	10, 19	sylancom 587	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
21		omex 9660	. . . . 5 ⊢ ω ∈ V
22		fnrndomg 10553	. . . . . . 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 9021	. . . . . 6 ⊢ ((ran 𝐹 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω)
25	23, 24	sylancom 587	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω)
26		xpdom2g 9086	. . . . 5 ⊢ ((ω ∈ V ∧ ran 𝐹 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
27	21, 25, 26	sylancr 586	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
28		domtr 9021	. . . 4 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ran 𝐹) ∧ (ω × ran 𝐹) ≼ (ω × ω)) → (𝐴 × ran 𝐹) ≼ (ω × ω))
29	20, 27, 28	syl2anc 583	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ω))
30		xpomen 10032	. . 3 ⊢ (ω × ω) ≈ ω
31		domentr 9027	. . 3 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ran 𝐹) ≼ ω)
32	29, 30, 31	sylancl 585	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ ω)
33		domtr 9021	. 2 ⊢ ((𝐹 ≼ (𝐴 × ran 𝐹) ∧ (𝐴 × ran 𝐹) ≼ ω) → 𝐹 ≼ ω)
34	18, 32, 33	syl2anc 583	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099  Vcvv 3470   ⊆ wss 3945   class class class wbr 5142   × cxp 5670  dom cdm 5672  ran crn 5673  Fun wfun 6536   Fn wfn 6537  ⟶wf 6538  ωcom 7864   ≈ cen 8954   ≼ cdom 8955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-ac2 10480

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-oi 9527  df-card 9956  df-acn 9959  df-ac 10133

This theorem is referenced by:  mptct  10555  mpocti  32491  mptctf  32493  omssubadd  33914