Theorem rn1st 45280

Description: The range of a function with a first-countable domain is itself first-countable. This is a variation of 1stcrestlem 23460, with a not-free hypothesis replacing a disjoint variable constraint. (Contributed by Glauco Siliprandi, 24-Jan-2025.)

Hypothesis

Ref	Expression
rn1st.1	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
rn1st	⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)

Proof of Theorem rn1st

Step	Hyp	Ref	Expression
1		ordom 7897	. . . . . 6 ⊢ Ord ω
2		reldom 8991	. . . . . . . 8 ⊢ Rel ≼
3	2	brrelex2i 5742	. . . . . . 7 ⊢ (𝐵 ≼ ω → ω ∈ V)
4		elong 6392	. . . . . . 7 ⊢ (ω ∈ V → (ω ∈ On ↔ Ord ω))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω))
6	1, 5	mpbiri 258	. . . . 5 ⊢ (𝐵 ≼ ω → ω ∈ On)
7		ondomen 10077	. . . . 5 ⊢ ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card)
8	6, 7	mpancom 688	. . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ dom card)
9		rn1st.1	. . . . 5 ⊢ Ⅎ𝑥𝐵
10		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
11	9, 10	dmmptssf 45237	. . . 4 ⊢ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵
12		ssnum 10079	. . . 4 ⊢ ((𝐵 ∈ dom card ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card)
13	8, 11, 12	sylancl 586	. . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card)
14		funmpt 6604	. . . 4 ⊢ Fun (𝑥 ∈ 𝐵 ↦ 𝐶)
15		funforn 6827	. . . 4 ⊢ (Fun (𝑥 ∈ 𝐵 ↦ 𝐶) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶))
16	14, 15	mpbi 230	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶)
17		fodomnum 10097	. . 3 ⊢ (dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card → ((𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶)))
18	13, 16, 17	mpisyl 21	. 2 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶))
19		ctex 9004	. . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V)
20		ssdomg 9040	. . . 4 ⊢ (𝐵 ∈ V → (dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵 → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵))
21	19, 11, 20	mpisyl 21	. . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵)
22		domtr 9047	. . 3 ⊢ ((dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵 ∧ 𝐵 ≼ ω) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)
23	21, 22	mpancom 688	. 2 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)
24		domtr 9047	. 2 ⊢ ((ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)
25	18, 23, 24	syl2anc 584	1 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2108  Ⅎwnfc 2890  Vcvv 3480   ⊆ wss 3951   class class class wbr 5143   ↦ cmpt 5225  dom cdm 5685  ran crn 5686  Ord word 6383  Oncon0 6384  Fun wfun 6555  –onto→wfo 6559  ωcom 7887   ≼ cdom 8983  cardccrd 9975

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-card 9979  df-acn 9982

This theorem is referenced by:  saliunclf  46337