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Theorem enumctlemm 7126
Description: Lemma for enumct 7127. 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 5450 . . . . . . 7 (𝐹:𝑁onto𝐴𝐹:𝑁𝐴)
31, 2syl 14 . . . . . 6 (𝜑𝐹:𝑁𝐴)
43ffvelcdmda 5664 . . . . 5 ((𝜑𝑘𝑁) → (𝐹𝑘) ∈ 𝐴)
54adantlr 477 . . . 4 (((𝜑𝑘 ∈ ω) ∧ 𝑘𝑁) → (𝐹𝑘) ∈ 𝐴)
6 enumctlemm.n0 . . . . . 6 (𝜑 → ∅ ∈ 𝑁)
73, 6ffvelcdmd 5665 . . . . 5 (𝜑 → (𝐹‘∅) ∈ 𝐴)
87ad2antrr 488 . . . 4 (((𝜑𝑘 ∈ ω) ∧ ¬ 𝑘𝑁) → (𝐹‘∅) ∈ 𝐴)
9 simpr 110 . . . . 5 ((𝜑𝑘 ∈ ω) → 𝑘 ∈ ω)
10 enumctlemm.n . . . . . 6 (𝜑𝑁 ∈ ω)
1110adantr 276 . . . . 5 ((𝜑𝑘 ∈ ω) → 𝑁 ∈ ω)
12 nndcel 6514 . . . . 5 ((𝑘 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑘𝑁)
139, 11, 12syl2anc 411 . . . 4 ((𝜑𝑘 ∈ ω) → DECID 𝑘𝑁)
145, 8, 13ifcldadc 3575 . . 3 ((𝜑𝑘 ∈ ω) → if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)) ∈ 𝐴)
15 enumctlemm.g . . 3 𝐺 = (𝑘 ∈ ω ↦ if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)))
1614, 15fmptd 5683 . 2 (𝜑𝐺:ω⟶𝐴)
17 foelrn 5766 . . . . . 6 ((𝐹:𝑁onto𝐴𝑦𝐴) → ∃𝑥𝑁 𝑦 = (𝐹𝑥))
181, 17sylan 283 . . . . 5 ((𝜑𝑦𝐴) → ∃𝑥𝑁 𝑦 = (𝐹𝑥))
19 eleq1w 2248 . . . . . . . . . . 11 (𝑘 = 𝑥 → (𝑘𝑁𝑥𝑁))
20 fveq2 5527 . . . . . . . . . . 11 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
2119, 20ifbieq1d 3568 . . . . . . . . . 10 (𝑘 = 𝑥 → if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)) = if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)))
22 simpr 110 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → 𝑥𝑁)
2310adantr 276 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → 𝑁 ∈ ω)
24 elnn 4617 . . . . . . . . . . 11 ((𝑥𝑁𝑁 ∈ ω) → 𝑥 ∈ ω)
2522, 23, 24syl2anc 411 . . . . . . . . . 10 ((𝜑𝑥𝑁) → 𝑥 ∈ ω)
2622iftrued 3553 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)) = (𝐹𝑥))
273ffvelcdmda 5664 . . . . . . . . . . 11 ((𝜑𝑥𝑁) → (𝐹𝑥) ∈ 𝐴)
2826, 27eqeltrd 2264 . . . . . . . . . 10 ((𝜑𝑥𝑁) → if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)) ∈ 𝐴)
2915, 21, 25, 28fvmptd3 5622 . . . . . . . . 9 ((𝜑𝑥𝑁) → (𝐺𝑥) = if(𝑥𝑁, (𝐹𝑥), (𝐹‘∅)))
3029, 26eqtrd 2220 . . . . . . . 8 ((𝜑𝑥𝑁) → (𝐺𝑥) = (𝐹𝑥))
3130eqeq2d 2199 . . . . . . 7 ((𝜑𝑥𝑁) → (𝑦 = (𝐺𝑥) ↔ 𝑦 = (𝐹𝑥)))
3231rexbidva 2484 . . . . . 6 (𝜑 → (∃𝑥𝑁 𝑦 = (𝐺𝑥) ↔ ∃𝑥𝑁 𝑦 = (𝐹𝑥)))
3332adantr 276 . . . . 5 ((𝜑𝑦𝐴) → (∃𝑥𝑁 𝑦 = (𝐺𝑥) ↔ ∃𝑥𝑁 𝑦 = (𝐹𝑥)))
3418, 33mpbird 167 . . . 4 ((𝜑𝑦𝐴) → ∃𝑥𝑁 𝑦 = (𝐺𝑥))
35 omelon 4620 . . . . . . 7 ω ∈ On
3635onelssi 4441 . . . . . 6 (𝑁 ∈ ω → 𝑁 ⊆ ω)
37 ssrexv 3232 . . . . . 6 (𝑁 ⊆ ω → (∃𝑥𝑁 𝑦 = (𝐺𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
3810, 36, 373syl 17 . . . . 5 (𝜑 → (∃𝑥𝑁 𝑦 = (𝐺𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
3938adantr 276 . . . 4 ((𝜑𝑦𝐴) → (∃𝑥𝑁 𝑦 = (𝐺𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
4034, 39mpd 13 . . 3 ((𝜑𝑦𝐴) → ∃𝑥 ∈ ω 𝑦 = (𝐺𝑥))
4140ralrimiva 2560 . 2 (𝜑 → ∀𝑦𝐴𝑥 ∈ ω 𝑦 = (𝐺𝑥))
42 dffo3 5676 . 2 (𝐺:ω–onto𝐴 ↔ (𝐺:ω⟶𝐴 ∧ ∀𝑦𝐴𝑥 ∈ ω 𝑦 = (𝐺𝑥)))
4316, 41, 42sylanbrc 417 1 (𝜑𝐺:ω–onto𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  DECID wdc 835   = wceq 1363  wcel 2158  wral 2465  wrex 2466  wss 3141  c0 3434  ifcif 3546  cmpt 4076  ωcom 4601  wf 5224  ontowfo 5226  cfv 5228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fo 5234  df-fv 5236
This theorem is referenced by:  enumct  7127
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