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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 𝑥𝐵
Assertion
Ref Expression
rn1st (𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)

Proof of Theorem rn1st
StepHypRef Expression
1 ordom 7897 . . . . . 6 Ord ω
2 reldom 8991 . . . . . . . 8 Rel ≼
32brrelex2i 5742 . . . . . . 7 (𝐵 ≼ ω → ω ∈ V)
4 elong 6392 . . . . . . 7 (ω ∈ V → (ω ∈ On ↔ Ord ω))
53, 4syl 17 . . . . . 6 (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω))
61, 5mpbiri 258 . . . . 5 (𝐵 ≼ ω → ω ∈ On)
7 ondomen 10077 . . . . 5 ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card)
86, 7mpancom 688 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ dom card)
9 rn1st.1 . . . . 5 𝑥𝐵
10 eqid 2737 . . . . 5 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
119, 10dmmptssf 45237 . . . 4 dom (𝑥𝐵𝐶) ⊆ 𝐵
12 ssnum 10079 . . . 4 ((𝐵 ∈ dom card ∧ dom (𝑥𝐵𝐶) ⊆ 𝐵) → dom (𝑥𝐵𝐶) ∈ dom card)
138, 11, 12sylancl 586 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ∈ dom card)
14 funmpt 6604 . . . 4 Fun (𝑥𝐵𝐶)
15 funforn 6827 . . . 4 (Fun (𝑥𝐵𝐶) ↔ (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶))
1614, 15mpbi 230 . . 3 (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶)
17 fodomnum 10097 . . 3 (dom (𝑥𝐵𝐶) ∈ dom card → ((𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶) → ran (𝑥𝐵𝐶) ≼ dom (𝑥𝐵𝐶)))
1813, 16, 17mpisyl 21 . 2 (𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ dom (𝑥𝐵𝐶))
19 ctex 9004 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ V)
20 ssdomg 9040 . . . 4 (𝐵 ∈ V → (dom (𝑥𝐵𝐶) ⊆ 𝐵 → dom (𝑥𝐵𝐶) ≼ 𝐵))
2119, 11, 20mpisyl 21 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ 𝐵)
22 domtr 9047 . . 3 ((dom (𝑥𝐵𝐶) ≼ 𝐵𝐵 ≼ ω) → dom (𝑥𝐵𝐶) ≼ ω)
2321, 22mpancom 688 . 2 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ ω)
24 domtr 9047 . 2 ((ran (𝑥𝐵𝐶) ≼ dom (𝑥𝐵𝐶) ∧ dom (𝑥𝐵𝐶) ≼ ω) → ran (𝑥𝐵𝐶) ≼ ω)
2518, 23, 24syl2anc 584 1 (𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wnfc 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  ontowfo 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
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