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Theorem mnurndlem2 43343
Description: Lemma for mnurnd 43344. 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 479 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑈𝑀)
53adantr 479 . . . . . . . 8 ((𝜑𝑎𝐴) → 𝐴𝑈)
6 simpr 483 . . . . . . . 8 ((𝜑𝑎𝐴) → 𝑎𝐴)
71, 4, 5, 6mnutrcld 43340 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝑈)
8 mnurndlem2.4 . . . . . . . . 9 (𝜑𝐹:𝐴𝑈)
98ffvelcdmda 7085 . . . . . . . 8 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ 𝑈)
101, 4, 9, 5mnuprd 43337 . . . . . . 7 ((𝜑𝑎𝐴) → {(𝐹𝑎), 𝐴} ∈ 𝑈)
111, 4, 7, 10mnuprd 43337 . . . . . 6 ((𝜑𝑎𝐴) → {𝑎, {(𝐹𝑎), 𝐴}} ∈ 𝑈)
1211ralrimiva 3144 . . . . 5 (𝜑 → ∀𝑎𝐴 {𝑎, {(𝐹𝑎), 𝐴}} ∈ 𝑈)
13 eqid 2730 . . . . . 6 (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}}) = (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})
1413rnmptss 7123 . . . . 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 43332 . . 3 (𝜑 → ∃𝑤𝑈𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))
17 simprl 767 . . . 4 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → 𝑤𝑈)
18 sseq2 4007 . . . . 5 (𝑏 = 𝑤 → (ran 𝐹𝑏 ↔ ran 𝐹𝑤))
1918adantl 480 . . . 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 479 . . . . 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 769 . . . . 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 43342 . . . 4 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → ran 𝐹𝑤)
2417, 19, 23rspcedvd 3613 . . 3 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → ∃𝑏𝑈 ran 𝐹𝑏)
2516, 24rexlimddv 3159 . 2 (𝜑 → ∃𝑏𝑈 ran 𝐹𝑏)
261, 2, 25mnuss2d 43325 1 (𝜑 → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1537   = wceq 1539  wcel 2104  {cab 2707  wral 3059  wrex 3068  Vcvv 3472  wss 3947  𝒫 cpw 4601  {cpr 4629   cuni 4907  cmpt 5230  ran crn 5676  wf 6538  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-reg 9589
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-eprel 5579  df-fr 5630  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550
This theorem is referenced by:  mnurnd  43344
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