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Theorem smfpimcclem 41677
Description: Lemma for smfpimcc 41678 given the choice function 𝐶. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smfpimcclem.n 𝑛𝜑
smfpimcclem.z 𝑍𝑉
smfpimcclem.s (𝜑𝑆𝑊)
smfpimcclem.c ((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) → (𝐶𝑦) ∈ 𝑦)
smfpimcclem.h 𝐻 = (𝑛𝑍 ↦ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
Assertion
Ref Expression
smfpimcclem (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
Distinct variable groups:   𝐴,   𝐴,𝑠,𝑦   𝐶,𝑠,𝑦   ,𝐹   𝐹,𝑠,𝑦   ,𝐻   𝑆,,𝑛   𝑆,𝑠,𝑦,𝑛   ,𝑍,𝑛   𝑦,𝑍   𝜑,𝑦
Allowed substitution hints:   𝜑(,𝑛,𝑠)   𝐴(𝑛)   𝐶(,𝑛)   𝐹(𝑛)   𝐻(𝑦,𝑛,𝑠)   𝑉(𝑦,,𝑛,𝑠)   𝑊(𝑦,,𝑛,𝑠)   𝑍(𝑠)

Proof of Theorem smfpimcclem
StepHypRef Expression
1 smfpimcclem.n . . 3 𝑛𝜑
2 nfcv 2907 . . . . 5 𝑠𝑆
32ssrab2f 39974 . . . 4 {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ⊆ 𝑆
4 eqid 2765 . . . . . . 7 {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} = {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}
5 smfpimcclem.s . . . . . . 7 (𝜑𝑆𝑊)
64, 5rabexd 4976 . . . . . 6 (𝜑 → {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ V)
76adantr 472 . . . . 5 ((𝜑𝑛𝑍) → {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ V)
8 simpl 474 . . . . . 6 ((𝜑𝑛𝑍) → 𝜑)
9 simpr 477 . . . . . . 7 ((𝜑𝑛𝑍) → 𝑛𝑍)
10 eqid 2765 . . . . . . . 8 (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) = (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})
1110elrnmpt1 5545 . . . . . . 7 ((𝑛𝑍 ∧ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ V) → {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
129, 7, 11syl2anc 579 . . . . . 6 ((𝜑𝑛𝑍) → {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
138, 12jca 507 . . . . 5 ((𝜑𝑛𝑍) → (𝜑 ∧ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})))
14 eleq1 2832 . . . . . . . 8 (𝑦 = {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} → (𝑦 ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ↔ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})))
1514anbi2d 622 . . . . . . 7 (𝑦 = {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} → ((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) ↔ (𝜑 ∧ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))))
16 fveq2 6379 . . . . . . . 8 (𝑦 = {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} → (𝐶𝑦) = (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
17 id 22 . . . . . . . 8 (𝑦 = {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} → 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})
1816, 17eleq12d 2838 . . . . . . 7 (𝑦 = {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} → ((𝐶𝑦) ∈ 𝑦 ↔ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
1915, 18imbi12d 335 . . . . . 6 (𝑦 = {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} → (((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) → (𝐶𝑦) ∈ 𝑦) ↔ ((𝜑 ∧ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) → (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})))
20 smfpimcclem.c . . . . . 6 ((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) → (𝐶𝑦) ∈ 𝑦)
2119, 20vtoclg 3418 . . . . 5 ({𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ V → ((𝜑 ∧ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) → (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
227, 13, 21sylc 65 . . . 4 ((𝜑𝑛𝑍) → (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})
233, 22sseldi 3761 . . 3 ((𝜑𝑛𝑍) → (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ 𝑆)
24 smfpimcclem.h . . 3 𝐻 = (𝑛𝑍 ↦ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
251, 23, 24fmptdf 6581 . 2 (𝜑𝐻:𝑍𝑆)
26 nfcv 2907 . . . . . . . . 9 𝑠𝐶
27 nfrab1 3270 . . . . . . . . 9 𝑠{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}
2826, 27nffv 6389 . . . . . . . 8 𝑠(𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})
29 nfcv 2907 . . . . . . . . 9 𝑠((𝐹𝑛) “ 𝐴)
30 nfcv 2907 . . . . . . . . . 10 𝑠dom (𝐹𝑛)
3128, 30nfin 3982 . . . . . . . . 9 𝑠((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛))
3229, 31nfeq 2919 . . . . . . . 8 𝑠((𝐹𝑛) “ 𝐴) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛))
33 ineq1 3971 . . . . . . . . 9 (𝑠 = (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) → (𝑠 ∩ dom (𝐹𝑛)) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛)))
3433eqeq2d 2775 . . . . . . . 8 (𝑠 = (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) → (((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛)) ↔ ((𝐹𝑛) “ 𝐴) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛))))
3528, 2, 32, 34elrabf 3517 . . . . . . 7 ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ↔ ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ 𝑆 ∧ ((𝐹𝑛) “ 𝐴) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛))))
3622, 35sylib 209 . . . . . 6 ((𝜑𝑛𝑍) → ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ 𝑆 ∧ ((𝐹𝑛) “ 𝐴) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛))))
3736simprd 489 . . . . 5 ((𝜑𝑛𝑍) → ((𝐹𝑛) “ 𝐴) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛)))
3824a1i 11 . . . . . . 7 (𝜑𝐻 = (𝑛𝑍 ↦ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})))
3922elexd 3367 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ V)
4038, 39fvmpt2d 6486 . . . . . 6 ((𝜑𝑛𝑍) → (𝐻𝑛) = (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
4140ineq1d 3977 . . . . 5 ((𝜑𝑛𝑍) → ((𝐻𝑛) ∩ dom (𝐹𝑛)) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛)))
4237, 41eqtr4d 2802 . . . 4 ((𝜑𝑛𝑍) → ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))
4342ex 401 . . 3 (𝜑 → (𝑛𝑍 → ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛))))
441, 43ralrimi 3104 . 2 (𝜑 → ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))
45 smfpimcclem.z . . . . . 6 𝑍𝑉
4645elexi 3366 . . . . 5 𝑍 ∈ V
4746mptex 6683 . . . 4 (𝑛𝑍 ↦ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) ∈ V
4824, 47eqeltri 2840 . . 3 𝐻 ∈ V
49 feq1 6206 . . . 4 ( = 𝐻 → (:𝑍𝑆𝐻:𝑍𝑆))
50 nfcv 2907 . . . . . 6 𝑛
51 nfmpt1 4908 . . . . . . 7 𝑛(𝑛𝑍 ↦ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
5224, 51nfcxfr 2905 . . . . . 6 𝑛𝐻
5350, 52nfeq 2919 . . . . 5 𝑛 = 𝐻
54 fveq1 6378 . . . . . . 7 ( = 𝐻 → (𝑛) = (𝐻𝑛))
5554ineq1d 3977 . . . . . 6 ( = 𝐻 → ((𝑛) ∩ dom (𝐹𝑛)) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))
5655eqeq2d 2775 . . . . 5 ( = 𝐻 → (((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛)) ↔ ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛))))
5753, 56ralbid 3130 . . . 4 ( = 𝐻 → (∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛)) ↔ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛))))
5849, 57anbi12d 624 . . 3 ( = 𝐻 → ((:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))) ↔ (𝐻:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))))
5948, 58spcev 3453 . 2 ((𝐻:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛))) → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
6025, 44, 59syl2anc 579 1 (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wex 1874  wnf 1878  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  cin 3733  cmpt 4890  ccnv 5278  dom cdm 5279  ran crn 5280  cima 5282  wf 6066  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078
This theorem is referenced by:  smfpimcc  41678
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