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Theorem mnurndlem2 44306
Description: Lemma for mnurnd 44307. Deduction theorem input. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnurndlem2.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnurndlem2.2 (𝜑𝑈𝑀)
mnurndlem2.3 (𝜑𝐴𝑈)
mnurndlem2.4 (𝜑𝐹:𝐴𝑈)
mnurndlem2.5 𝐴 ∈ V
Assertion
Ref Expression
mnurndlem2 (𝜑 → ran 𝐹𝑈)
Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙   𝑈,𝑟,𝑘,𝑚,𝑛,𝑝,𝑙
Allowed substitution hints:   𝜑(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnurndlem2
Dummy variables 𝑣 𝑎 𝑏 𝑤 𝑖 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnurndlem2.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnurndlem2.2 . 2 (𝜑𝑈𝑀)
3 mnurndlem2.3 . . . 4 (𝜑𝐴𝑈)
42adantr 480 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑈𝑀)
53adantr 480 . . . . . . . 8 ((𝜑𝑎𝐴) → 𝐴𝑈)
6 simpr 484 . . . . . . . 8 ((𝜑𝑎𝐴) → 𝑎𝐴)
71, 4, 5, 6mnutrcld 44303 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝑈)
8 mnurndlem2.4 . . . . . . . . 9 (𝜑𝐹:𝐴𝑈)
98ffvelcdmda 7103 . . . . . . . 8 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ 𝑈)
101, 4, 9, 5mnuprd 44300 . . . . . . 7 ((𝜑𝑎𝐴) → {(𝐹𝑎), 𝐴} ∈ 𝑈)
111, 4, 7, 10mnuprd 44300 . . . . . 6 ((𝜑𝑎𝐴) → {𝑎, {(𝐹𝑎), 𝐴}} ∈ 𝑈)
1211ralrimiva 3145 . . . . 5 (𝜑 → ∀𝑎𝐴 {𝑎, {(𝐹𝑎), 𝐴}} ∈ 𝑈)
13 eqid 2736 . . . . . 6 (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}}) = (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})
1413rnmptss 7142 . . . . 5 (∀𝑎𝐴 {𝑎, {(𝐹𝑎), 𝐴}} ∈ 𝑈 → ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}}) ⊆ 𝑈)
1512, 14syl 17 . . . 4 (𝜑 → ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}}) ⊆ 𝑈)
161, 2, 3, 15mnuop3d 44295 . . 3 (𝜑 → ∃𝑤𝑈𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))
17 simprl 770 . . . 4 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → 𝑤𝑈)
18 sseq2 4009 . . . . 5 (𝑏 = 𝑤 → (ran 𝐹𝑏 ↔ ran 𝐹𝑤))
1918adantl 481 . . . 4 (((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) ∧ 𝑏 = 𝑤) → (ran 𝐹𝑏 ↔ ran 𝐹𝑤))
208adantr 480 . . . . 5 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → 𝐹:𝐴𝑈)
21 mnurndlem2.5 . . . . 5 𝐴 ∈ V
22 simprr 772 . . . . 5 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))
2320, 21, 22mnurndlem1 44305 . . . 4 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → ran 𝐹𝑤)
2417, 19, 23rspcedvd 3623 . . 3 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → ∃𝑏𝑈 ran 𝐹𝑏)
2516, 24rexlimddv 3160 . 2 (𝜑 → ∃𝑏𝑈 ran 𝐹𝑏)
261, 2, 25mnuss2d 44288 1 (𝜑 → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wcel 2107  {cab 2713  wral 3060  wrex 3069  Vcvv 3479  wss 3950  𝒫 cpw 4599  {cpr 4627   cuni 4906  cmpt 5224  ran crn 5685  wf 6556  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-reg 9633
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-eprel 5583  df-fr 5636  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568
This theorem is referenced by:  mnurnd  44307
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