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Theorem ctiunctlemf 11857
Description: Lemma for ctiunct 11859. (Contributed by Jim Kingdon, 28-Oct-2023.)
Hypotheses
Ref Expression
ctiunct.som (𝜑𝑆 ⊆ ω)
ctiunct.sdc (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
ctiunct.f (𝜑𝐹:𝑆onto𝐴)
ctiunct.tom ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)
ctiunct.tdc ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)
ctiunct.g ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)
ctiunct.j (𝜑𝐽:ω–1-1-onto→(ω × ω))
ctiunct.u 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}
ctiunct.h 𝐻 = (𝑛𝑈 ↦ ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))))
Assertion
Ref Expression
ctiunctlemf (𝜑𝐻:𝑈 𝑥𝐴 𝐵)
Distinct variable groups:   𝐴,𝑛,𝑥   𝐵,𝑛   𝑥,𝐹,𝑧   𝑥,𝐽,𝑧   𝑧,𝑆   𝑧,𝑇   𝑈,𝑛   𝜑,𝑛,𝑥   𝑥,𝑧,𝑛
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)   𝐵(𝑥,𝑧)   𝑆(𝑥,𝑛)   𝑇(𝑥,𝑛)   𝑈(𝑥,𝑧)   𝐹(𝑛)   𝐺(𝑥,𝑧,𝑛)   𝐻(𝑥,𝑧,𝑛)   𝐽(𝑛)

Proof of Theorem ctiunctlemf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ctiunct.f . . . . . . . 8 (𝜑𝐹:𝑆onto𝐴)
21adantr 272 . . . . . . 7 ((𝜑𝑛𝑈) → 𝐹:𝑆onto𝐴)
3 fof 5313 . . . . . . 7 (𝐹:𝑆onto𝐴𝐹:𝑆𝐴)
42, 3syl 14 . . . . . 6 ((𝜑𝑛𝑈) → 𝐹:𝑆𝐴)
5 ctiunct.som . . . . . . . 8 (𝜑𝑆 ⊆ ω)
65adantr 272 . . . . . . 7 ((𝜑𝑛𝑈) → 𝑆 ⊆ ω)
7 ctiunct.sdc . . . . . . . 8 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
87adantr 272 . . . . . . 7 ((𝜑𝑛𝑈) → ∀𝑛 ∈ ω DECID 𝑛𝑆)
9 ctiunct.tom . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)
109adantlr 466 . . . . . . 7 (((𝜑𝑛𝑈) ∧ 𝑥𝐴) → 𝑇 ⊆ ω)
11 ctiunct.tdc . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)
1211adantlr 466 . . . . . . 7 (((𝜑𝑛𝑈) ∧ 𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)
13 ctiunct.g . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)
1413adantlr 466 . . . . . . 7 (((𝜑𝑛𝑈) ∧ 𝑥𝐴) → 𝐺:𝑇onto𝐵)
15 ctiunct.j . . . . . . . 8 (𝜑𝐽:ω–1-1-onto→(ω × ω))
1615adantr 272 . . . . . . 7 ((𝜑𝑛𝑈) → 𝐽:ω–1-1-onto→(ω × ω))
17 ctiunct.u . . . . . . 7 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}
18 simpr 109 . . . . . . 7 ((𝜑𝑛𝑈) → 𝑛𝑈)
196, 8, 2, 10, 12, 14, 16, 17, 18ctiunctlemu1st 11853 . . . . . 6 ((𝜑𝑛𝑈) → (1st ‘(𝐽𝑛)) ∈ 𝑆)
204, 19ffvelrnd 5522 . . . . 5 ((𝜑𝑛𝑈) → (𝐹‘(1st ‘(𝐽𝑛))) ∈ 𝐴)
21 fof 5313 . . . . . . . . . . 11 (𝐺:𝑇onto𝐵𝐺:𝑇𝐵)
2213, 21syl 14 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐺:𝑇𝐵)
2322ralrimiva 2480 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐺:𝑇𝐵)
2423adantr 272 . . . . . . . 8 ((𝜑𝑛𝑈) → ∀𝑥𝐴 𝐺:𝑇𝐵)
25 rspsbc 2961 . . . . . . . 8 ((𝐹‘(1st ‘(𝐽𝑛))) ∈ 𝐴 → (∀𝑥𝐴 𝐺:𝑇𝐵[(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥]𝐺:𝑇𝐵))
2620, 24, 25sylc 62 . . . . . . 7 ((𝜑𝑛𝑈) → [(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥]𝐺:𝑇𝐵)
27 sbcfg 5239 . . . . . . . 8 ((𝐹‘(1st ‘(𝐽𝑛))) ∈ 𝐴 → ([(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥]𝐺:𝑇𝐵(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺:(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝑇(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐵))
2820, 27syl 14 . . . . . . 7 ((𝜑𝑛𝑈) → ([(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥]𝐺:𝑇𝐵(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺:(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝑇(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐵))
2926, 28mpbid 146 . . . . . 6 ((𝜑𝑛𝑈) → (𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺:(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝑇(𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐵)
306, 8, 2, 10, 12, 14, 16, 17, 18ctiunctlemu2nd 11854 . . . . . 6 ((𝜑𝑛𝑈) → (2nd ‘(𝐽𝑛)) ∈ (𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝑇)
3129, 30ffvelrnd 5522 . . . . 5 ((𝜑𝑛𝑈) → ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ (𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐵)
32 csbeq1 2976 . . . . . . 7 (𝑦 = (𝐹‘(1st ‘(𝐽𝑛))) → 𝑦 / 𝑥𝐵 = (𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐵)
3332eleq2d 2185 . . . . . 6 (𝑦 = (𝐹‘(1st ‘(𝐽𝑛))) → (((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ 𝑦 / 𝑥𝐵 ↔ ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ (𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐵))
3433rspcev 2761 . . . . 5 (((𝐹‘(1st ‘(𝐽𝑛))) ∈ 𝐴 ∧ ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ (𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐵) → ∃𝑦𝐴 ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ 𝑦 / 𝑥𝐵)
3520, 31, 34syl2anc 406 . . . 4 ((𝜑𝑛𝑈) → ∃𝑦𝐴 ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ 𝑦 / 𝑥𝐵)
36 eliun 3785 . . . 4 (((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ 𝑦𝐴 𝑦 / 𝑥𝐵 ↔ ∃𝑦𝐴 ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ 𝑦 / 𝑥𝐵)
3735, 36sylibr 133 . . 3 ((𝜑𝑛𝑈) → ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ 𝑦𝐴 𝑦 / 𝑥𝐵)
38 nfcv 2256 . . . 4 𝑦𝐵
39 nfcsb1v 3003 . . . 4 𝑥𝑦 / 𝑥𝐵
40 csbeq1a 2981 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
4138, 39, 40cbviun 3818 . . 3 𝑥𝐴 𝐵 = 𝑦𝐴 𝑦 / 𝑥𝐵
4237, 41syl6eleqr 2209 . 2 ((𝜑𝑛𝑈) → ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) ∈ 𝑥𝐴 𝐵)
43 ctiunct.h . 2 𝐻 = (𝑛𝑈 ↦ ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))))
4442, 43fmptd 5540 1 (𝜑𝐻:𝑈 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  DECID wdc 802   = wceq 1314  wcel 1463  wral 2391  wrex 2392  {crab 2395  [wsbc 2880  csb 2973  wss 3039   ciun 3781  cmpt 3957  ωcom 4472   × cxp 4505  wf 5087  ontowfo 5089  1-1-ontowf1o 5090  cfv 5091  1st c1st 6002  2nd c2nd 6003
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fo 5097  df-fv 5099
This theorem is referenced by:  ctiunctlemfo  11858
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