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Theorem enumctlemm 7012
 Description: Lemma for enumct 7013. The case where 𝑁 is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)
Hypotheses
Ref Expression
enumctlemm.f (𝜑𝐹:𝑁onto𝐴)
enumctlemm.n (𝜑𝑁 ∈ ω)
enumctlemm.n0 (𝜑 → ∅ ∈ 𝑁)
enumctlemm.g 𝐺 = (𝑘 ∈ ω ↦ if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)))
Assertion
Ref Expression
enumctlemm (𝜑𝐺:ω–onto𝐴)
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑁   𝜑,𝑘
Allowed substitution hint:   𝐺(𝑘)

Proof of Theorem enumctlemm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enumctlemm.f . . . . . . 7 (𝜑𝐹:𝑁onto𝐴)
2 fof 5355 . . . . . . 7 (𝐹:𝑁onto𝐴𝐹:𝑁𝐴)
31, 2syl 14 . . . . . 6 (𝜑𝐹:𝑁𝐴)
43ffvelrnda 5565 . . . . 5 ((𝜑𝑘𝑁) → (𝐹𝑘) ∈ 𝐴)
54adantlr 469 . . . 4 (((𝜑𝑘 ∈ ω) ∧ 𝑘𝑁) → (𝐹𝑘) ∈ 𝐴)
6 enumctlemm.n0 . . . . . 6 (𝜑 → ∅ ∈ 𝑁)
73, 6ffvelrnd 5566 . . . . 5 (𝜑 → (𝐹‘∅) ∈ 𝐴)
87ad2antrr 480 . . . 4 (((𝜑𝑘 ∈ ω) ∧ ¬ 𝑘𝑁) → (𝐹‘∅) ∈ 𝐴)
9 simpr 109 . . . . 5 ((𝜑𝑘 ∈ ω) → 𝑘 ∈ ω)
10 enumctlemm.n . . . . . 6 (𝜑𝑁 ∈ ω)
1110adantr 274 . . . . 5 ((𝜑𝑘 ∈ ω) → 𝑁 ∈ ω)
12 nndcel 6406 . . . . 5 ((𝑘 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑘𝑁)
139, 11, 12syl2anc 409 . . . 4 ((𝜑𝑘 ∈ ω) → DECID 𝑘𝑁)
145, 8, 13ifcldadc 3507 . . 3 ((𝜑𝑘 ∈ ω) → if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)) ∈ 𝐴)
15 enumctlemm.g . . 3 𝐺 = (𝑘 ∈ ω ↦ if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)))
1614, 15fmptd 5584 . 2 (𝜑𝐺:ω⟶𝐴)
17 foelrn 5664 . . . . . 6 ((𝐹:𝑁onto𝐴𝑦𝐴) → ∃𝑥𝑁 𝑦 = (𝐹𝑥))
181, 17sylan 281 . . . . 5 ((𝜑𝑦𝐴) → ∃𝑥𝑁 𝑦 = (𝐹𝑥))
19 eleq1w 2201 . . . . . . . . . . 11 (𝑘 = 𝑥 → (𝑘𝑁𝑥𝑁))
20 fveq2 5431 . . . . . . . . . . 11 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
2119, 20ifbieq1d 3500 . . . . . . . . . 10 (𝑘 = 𝑥 → if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)) = if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)))
22 simpr 109 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → 𝑥𝑁)
2310adantr 274 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → 𝑁 ∈ ω)
24 elnn 4529 . . . . . . . . . . 11 ((𝑥𝑁𝑁 ∈ ω) → 𝑥 ∈ ω)
2522, 23, 24syl2anc 409 . . . . . . . . . 10 ((𝜑𝑥𝑁) → 𝑥 ∈ ω)
2622iftrued 3487 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)) = (𝐹𝑥))
273ffvelrnda 5565 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → (𝐹𝑥) ∈ 𝐴)
2826, 27eqeltrd 2217 . . . . . . . . . 10 ((𝜑𝑥𝑁) → if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)) ∈ 𝐴)
2915, 21, 25, 28fvmptd3 5524 . . . . . . . . 9 ((𝜑𝑥𝑁) → (𝐺𝑥) = if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)))
3029, 26eqtrd 2173 . . . . . . . 8 ((𝜑𝑥𝑁) → (𝐺𝑥) = (𝐹𝑥))
3130eqeq2d 2152 . . . . . . 7 ((𝜑𝑥𝑁) → (𝑦 = (𝐺𝑥) ↔ 𝑦 = (𝐹𝑥)))
3231rexbidva 2436 . . . . . 6 (𝜑 → (∃𝑥𝑁 𝑦 = (𝐺𝑥) ↔ ∃𝑥𝑁 𝑦 = (𝐹𝑥)))
3332adantr 274 . . . . 5 ((𝜑𝑦𝐴) → (∃𝑥𝑁 𝑦 = (𝐺𝑥) ↔ ∃𝑥𝑁 𝑦 = (𝐹𝑥)))
3418, 33mpbird 166 . . . 4 ((𝜑𝑦𝐴) → ∃𝑥𝑁 𝑦 = (𝐺𝑥))
35 omelon 4532 . . . . . . 7 ω ∈ On
3635onelssi 4360 . . . . . 6 (𝑁 ∈ ω → 𝑁 ⊆ ω)
37 ssrexv 3168 . . . . . 6 (𝑁 ⊆ ω → (∃𝑥𝑁 𝑦 = (𝐺𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
3810, 36, 373syl 17 . . . . 5 (𝜑 → (∃𝑥𝑁 𝑦 = (𝐺𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
3938adantr 274 . . . 4 ((𝜑𝑦𝐴) → (∃𝑥𝑁 𝑦 = (𝐺𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
4034, 39mpd 13 . . 3 ((𝜑𝑦𝐴) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥))
4140ralrimiva 2509 . 2 (𝜑 → ∀𝑦𝐴𝑥 ∈ ω 𝑦 = (𝐺𝑥))
42 dffo3 5577 . 2 (𝐺:ω–onto𝐴 ↔ (𝐺:ω⟶𝐴 ∧ ∀𝑦𝐴𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
4316, 41, 42sylanbrc 414 1 (𝜑𝐺:ω–onto𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 820   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418   ⊆ wss 3077  ∅c0 3369  ifcif 3480   ↦ cmpt 3998  ωcom 4513  ⟶wf 5129  –onto→wfo 5131  ‘cfv 5133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-iinf 4511 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-if 3481  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-id 4224  df-iord 4297  df-on 4299  df-suc 4302  df-iom 4514  df-xp 4555  df-rel 4556  df-cnv 4557  df-co 4558  df-dm 4559  df-rn 4560  df-res 4561  df-ima 4562  df-iota 5098  df-fun 5135  df-fn 5136  df-f 5137  df-fo 5139  df-fv 5141 This theorem is referenced by:  enumct  7013
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