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Theorem rn1st 43401
Description: The range of a function with a first-countable domain is itself first-countable. This is a variation of 1stcrestlem 22754, 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 7804 . . . . . 6 Ord ω
2 reldom 8847 . . . . . . . 8 Rel ≼
32brrelex2i 5687 . . . . . . 7 (𝐵 ≼ ω → ω ∈ V)
4 elong 6323 . . . . . . 7 (ω ∈ V → (ω ∈ On ↔ Ord ω))
53, 4syl 17 . . . . . 6 (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω))
61, 5mpbiri 257 . . . . 5 (𝐵 ≼ ω → ω ∈ On)
7 ondomen 9931 . . . . 5 ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card)
86, 7mpancom 686 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ dom card)
9 rn1st.1 . . . . 5 𝑥𝐵
10 eqid 2737 . . . . 5 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
119, 10dmmptssf 43355 . . . 4 dom (𝑥𝐵𝐶) ⊆ 𝐵
12 ssnum 9933 . . . 4 ((𝐵 ∈ dom card ∧ dom (𝑥𝐵𝐶) ⊆ 𝐵) → dom (𝑥𝐵𝐶) ∈ dom card)
138, 11, 12sylancl 586 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ∈ dom card)
14 funmpt 6536 . . . 4 Fun (𝑥𝐵𝐶)
15 funforn 6760 . . . 4 (Fun (𝑥𝐵𝐶) ↔ (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶))
1614, 15mpbi 229 . . 3 (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶)
17 fodomnum 9951 . . 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 8861 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ V)
20 ssdomg 8898 . . . 4 (𝐵 ∈ V → (dom (𝑥𝐵𝐶) ⊆ 𝐵 → dom (𝑥𝐵𝐶) ≼ 𝐵))
2119, 11, 20mpisyl 21 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ 𝐵)
22 domtr 8905 . . 3 ((dom (𝑥𝐵𝐶) ≼ 𝐵𝐵 ≼ ω) → dom (𝑥𝐵𝐶) ≼ ω)
2321, 22mpancom 686 . 2 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ ω)
24 domtr 8905 . 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 205  wcel 2106  wnfc 2885  Vcvv 3443  wss 3908   class class class wbr 5103  cmpt 5186  dom cdm 5631  ran crn 5632  Ord word 6314  Oncon0 6315  Fun wfun 6487  ontowfo 6491  ωcom 7794  cdom 8839  cardccrd 9829
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-er 8606  df-map 8725  df-en 8842  df-dom 8843  df-card 9833  df-acn 9836
This theorem is referenced by:  saliunclf  44457
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