Theorem fnct 10548

Description: If the domain of a function is countable, the function is countable. The proof uses fnrndomnum 10545 rather than fnrndomg 10546, and so does not require ax-ac 10465. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Revised by Vincent Gonzalez, 24-Aug-2026.)

Assertion

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

Proof of Theorem fnct

Step	Hyp	Ref	Expression
1		ctex 8973	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
2	1	adantl 487	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐴 ∈ V)
3		fndm 6639	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	eleq1d 2847	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
5	4	adantr 486	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
6	2, 5	mpbird 260	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → dom 𝐹 ∈ V)
7		fnfun 6636	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
8	7	adantr 486	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → Fun 𝐹)
9		funrnex 7955	. . . . 5 ⊢ (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
10	6, 8, 9	sylc 66	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ∈ V)
11	2, 10	xpexd 7754	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V)
12		dffn3 6719	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
13	12	birani 509	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹)
14		fssxp 6734	. . . 4 ⊢ (𝐹:𝐴⟶ran 𝐹 → 𝐹 ⊆ (𝐴 × ran 𝐹))
15	13, 14	syl 18	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ⊆ (𝐴 × ran 𝐹))
16		ssdomg 9010	. . 3 ⊢ ((𝐴 × ran 𝐹) ∈ V → (𝐹 ⊆ (𝐴 × ran 𝐹) → 𝐹 ≼ (𝐴 × ran 𝐹)))
17	11, 15, 16	sylc 66	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ (𝐴 × ran 𝐹))
18		xpdom1g 9076	. . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
19	10, 18	sylancom 600	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
20		omex 9626	. . . . 5 ⊢ ω ∈ V
21		omelon 9629	. . . . . . . . 9 ⊢ ω ∈ On
22	21	a1i 11	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ω ∈ On)
23		simpr 490	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐴 ≼ ω)
24		ondomen 10044	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝐴 ≼ ω) → 𝐴 ∈ dom card)
25	22, 23, 24	syl2anc 596	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐴 ∈ dom card)
26		simpl 488	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 Fn 𝐴)
27		fnrndomnum 10545	. . . . . . 7 ⊢ (𝐴 ∈ dom card → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴))
28	25, 26, 27	sylc 66	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ 𝐴)
29		domtr 9017	. . . . . 6 ⊢ ((ran 𝐹 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω)
30	28, 29	sylancom 600	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω)
31		xpdom2g 9075	. . . . 5 ⊢ ((ω ∈ V ∧ ran 𝐹 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
32	20, 30, 31	sylancr 599	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
33		domtr 9017	. . . 4 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ran 𝐹) ∧ (ω × ran 𝐹) ≼ (ω × ω)) → (𝐴 × ran 𝐹) ≼ (ω × ω))
34	19, 32, 33	syl2anc 596	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ω))
35		xpomen 10022	. . 3 ⊢ (ω × ω) ≈ ω
36		domentr 9023	. . 3 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ran 𝐹) ≼ ω)
37	34, 35, 36	sylancl 598	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ ω)
38		domtr 9017	. 2 ⊢ ((𝐹 ≼ (𝐴 × ran 𝐹) ∧ (𝐴 × ran 𝐹) ≼ ω) → 𝐹 ≼ ω)
39	17, 37, 38	syl2anc 596	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902   class class class wbr 5107   × cxp 5657  dom cdm 5659  ran crn 5660  Oncon0 6361  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ωcom 7866   ≈ cen 8953   ≼ cdom 8954  cardccrd 9944

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  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 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-oi 9486  df-card 9948  df-acn 9951

This theorem is used by:  mptct  10550  mpocti  33194  mptctf  33195  omssubadd  34819