Theorem rn1st 44277

Description: The range of a function with a first-countable domain is itself first-countable. This is a variation of 1stcrestlem 23176, 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 7867	. . . . . 6 ⊢ Ord ω
2		reldom 8947	. . . . . . . 8 ⊢ Rel ≼
3	2	brrelex2i 5733	. . . . . . 7 ⊢ (𝐵 ≼ ω → ω ∈ V)
4		elong 6372	. . . . . . 7 ⊢ (ω ∈ V → (ω ∈ On ↔ Ord ω))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω))
6	1, 5	mpbiri 257	. . . . 5 ⊢ (𝐵 ≼ ω → ω ∈ On)
7		ondomen 10034	. . . . 5 ⊢ ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card)
8	6, 7	mpancom 686	. . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ dom card)
9		rn1st.1	. . . . 5 ⊢ Ⅎ𝑥𝐵
10		eqid 2732	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
11	9, 10	dmmptssf 44233	. . . 4 ⊢ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵
12		ssnum 10036	. . . 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 6586	. . . 4 ⊢ Fun (𝑥 ∈ 𝐵 ↦ 𝐶)
15		funforn 6812	. . . 4 ⊢ (Fun (𝑥 ∈ 𝐵 ↦ 𝐶) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶))
16	14, 15	mpbi 229	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶)
17		fodomnum 10054	. . 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 8961	. . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V)
20		ssdomg 8998	. . . 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 9005	. . 3 ⊢ ((dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵 ∧ 𝐵 ≼ ω) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)
23	21, 22	mpancom 686	. 2 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)
24		domtr 9005	. 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 205   ∈ wcel 2106  Ⅎwnfc 2883  Vcvv 3474   ⊆ wss 3948   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677  Ord word 6363  Oncon0 6364  Fun wfun 6537  –onto→wfo 6541  ωcom 7857   ≼ cdom 8939  cardccrd 9932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-card 9936  df-acn 9939

This theorem is referenced by:  saliunclf  45337