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Theorem enumctlemm 6913
Description: Lemma for enumct 6914. 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 5281 . . . . . . 7 (𝐹:𝑁onto𝐴𝐹:𝑁𝐴)
31, 2syl 14 . . . . . 6 (𝜑𝐹:𝑁𝐴)
43ffvelrnda 5487 . . . . 5 ((𝜑𝑘𝑁) → (𝐹𝑘) ∈ 𝐴)
54adantlr 464 . . . 4 (((𝜑𝑘 ∈ ω) ∧ 𝑘𝑁) → (𝐹𝑘) ∈ 𝐴)
6 enumctlemm.n0 . . . . . 6 (𝜑 → ∅ ∈ 𝑁)
73, 6ffvelrnd 5488 . . . . 5 (𝜑 → (𝐹‘∅) ∈ 𝐴)
87ad2antrr 475 . . . 4 (((𝜑𝑘 ∈ ω) ∧ ¬ 𝑘𝑁) → (𝐹‘∅) ∈ 𝐴)
9 simpr 109 . . . . 5 ((𝜑𝑘 ∈ ω) → 𝑘 ∈ ω)
10 enumctlemm.n . . . . . 6 (𝜑𝑁 ∈ ω)
1110adantr 272 . . . . 5 ((𝜑𝑘 ∈ ω) → 𝑁 ∈ ω)
12 nndcel 6326 . . . . 5 ((𝑘 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑘𝑁)
139, 11, 12syl2anc 406 . . . 4 ((𝜑𝑘 ∈ ω) → DECID 𝑘𝑁)
145, 8, 13ifcldadc 3448 . . 3 ((𝜑𝑘 ∈ ω) → if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)) ∈ 𝐴)
15 enumctlemm.g . . 3 𝐺 = (𝑘 ∈ ω ↦ if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)))
1614, 15fmptd 5506 . 2 (𝜑𝐺:ω⟶𝐴)
17 foelrn 5586 . . . . . 6 ((𝐹:𝑁onto𝐴𝑦𝐴) → ∃𝑥𝑁 𝑦 = (𝐹𝑥))
181, 17sylan 279 . . . . 5 ((𝜑𝑦𝐴) → ∃𝑥𝑁 𝑦 = (𝐹𝑥))
19 eleq1w 2160 . . . . . . . . . . 11 (𝑘 = 𝑥 → (𝑘𝑁𝑥𝑁))
20 fveq2 5353 . . . . . . . . . . 11 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
2119, 20ifbieq1d 3441 . . . . . . . . . 10 (𝑘 = 𝑥 → if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)) = if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)))
22 simpr 109 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → 𝑥𝑁)
2310adantr 272 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → 𝑁 ∈ ω)
24 elnn 4457 . . . . . . . . . . 11 ((𝑥𝑁𝑁 ∈ ω) → 𝑥 ∈ ω)
2522, 23, 24syl2anc 406 . . . . . . . . . 10 ((𝜑𝑥𝑁) → 𝑥 ∈ ω)
2622iftrued 3428 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)) = (𝐹𝑥))
273ffvelrnda 5487 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → (𝐹𝑥) ∈ 𝐴)
2826, 27eqeltrd 2176 . . . . . . . . . 10 ((𝜑𝑥𝑁) → if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)) ∈ 𝐴)
2915, 21, 25, 28fvmptd3 5446 . . . . . . . . 9 ((𝜑𝑥𝑁) → (𝐺𝑥) = if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)))
3029, 26eqtrd 2132 . . . . . . . 8 ((𝜑𝑥𝑁) → (𝐺𝑥) = (𝐹𝑥))
3130eqeq2d 2111 . . . . . . 7 ((𝜑𝑥𝑁) → (𝑦 = (𝐺𝑥) ↔ 𝑦 = (𝐹𝑥)))
3231rexbidva 2393 . . . . . 6 (𝜑 → (∃𝑥𝑁 𝑦 = (𝐺𝑥) ↔ ∃𝑥𝑁 𝑦 = (𝐹𝑥)))
3332adantr 272 . . . . 5 ((𝜑𝑦𝐴) → (∃𝑥𝑁 𝑦 = (𝐺𝑥) ↔ ∃𝑥𝑁 𝑦 = (𝐹𝑥)))
3418, 33mpbird 166 . . . 4 ((𝜑𝑦𝐴) → ∃𝑥𝑁 𝑦 = (𝐺𝑥))
35 omelon 4460 . . . . . . 7 ω ∈ On
3635onelssi 4289 . . . . . 6 (𝑁 ∈ ω → 𝑁 ⊆ ω)
37 ssrexv 3109 . . . . . 6 (𝑁 ⊆ ω → (∃𝑥𝑁 𝑦 = (𝐺𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
3810, 36, 373syl 17 . . . . 5 (𝜑 → (∃𝑥𝑁 𝑦 = (𝐺𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
3938adantr 272 . . . 4 ((𝜑𝑦𝐴) → (∃𝑥𝑁 𝑦 = (𝐺𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
4034, 39mpd 13 . . 3 ((𝜑𝑦𝐴) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥))
4140ralrimiva 2464 . 2 (𝜑 → ∀𝑦𝐴𝑥 ∈ ω 𝑦 = (𝐺𝑥))
42 dffo3 5499 . 2 (𝐺:ω–onto𝐴 ↔ (𝐺:ω⟶𝐴 ∧ ∀𝑦𝐴𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
4316, 41, 42sylanbrc 411 1 (𝜑𝐺:ω–onto𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  DECID wdc 786   = wceq 1299  wcel 1448  wral 2375  wrex 2376  wss 3021  c0 3310  ifcif 3421  cmpt 3929  ωcom 4442  wf 5055  ontowfo 5057  cfv 5059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fo 5065  df-fv 5067
This theorem is referenced by:  enumct  6914
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