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Theorem mnurndlem2 44278
Description: Lemma for mnurnd 44279. 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 44275 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝑈)
8 mnurndlem2.4 . . . . . . . . 9 (𝜑𝐹:𝐴𝑈)
98ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ 𝑈)
101, 4, 9, 5mnuprd 44272 . . . . . . 7 ((𝜑𝑎𝐴) → {(𝐹𝑎), 𝐴} ∈ 𝑈)
111, 4, 7, 10mnuprd 44272 . . . . . 6 ((𝜑𝑎𝐴) → {𝑎, {(𝐹𝑎), 𝐴}} ∈ 𝑈)
1211ralrimiva 3144 . . . . 5 (𝜑 → ∀𝑎𝐴 {𝑎, {(𝐹𝑎), 𝐴}} ∈ 𝑈)
13 eqid 2735 . . . . . 6 (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}}) = (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})
1413rnmptss 7143 . . . . 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 44267 . . 3 (𝜑 → ∃𝑤𝑈𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))
17 simprl 771 . . . 4 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → 𝑤𝑈)
18 sseq2 4022 . . . . 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 773 . . . . 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 44277 . . . 4 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))) → ran 𝐹𝑤)
2417, 19, 23rspcedvd 3624 . . 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 44260 1 (𝜑 → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  wss 3963  𝒫 cpw 4605  {cpr 4633   cuni 4912  cmpt 5231  ran crn 5690  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-fr 5641  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  mnurnd  44279
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