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Theorem smfpimcclem 41530
Description: Lemma for smfpimcc 41531 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 2913 . . . . 5 𝑠𝑆
32ssrab2f 39821 . . . 4 {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ⊆ 𝑆
4 eqid 2771 . . . . . . 7 {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} = {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}
5 smfpimcclem.s . . . . . . 7 (𝜑𝑆𝑊)
64, 5rabexd 4948 . . . . . 6 (𝜑 → {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ V)
76adantr 466 . . . . 5 ((𝜑𝑛𝑍) → {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ V)
8 simpl 468 . . . . . 6 ((𝜑𝑛𝑍) → 𝜑)
9 simpr 471 . . . . . . 7 ((𝜑𝑛𝑍) → 𝑛𝑍)
10 eqid 2771 . . . . . . . 8 (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) = (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})
1110elrnmpt1 5511 . . . . . . 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 573 . . . . . 6 ((𝜑𝑛𝑍) → {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
138, 12jca 501 . . . . 5 ((𝜑𝑛𝑍) → (𝜑 ∧ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))} ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})))
14 eleq1 2838 . . . . . . . 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 614 . . . . . . 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 6333 . . . . . . . 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 2844 . . . . . . 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 333 . . . . . 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 3417 . . . . 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 3750 . . 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 6532 . 2 (𝜑𝐻:𝑍𝑆)
26 nfcv 2913 . . . . . . . . 9 𝑠𝐶
27 nfrab1 3271 . . . . . . . . 9 𝑠{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}
2826, 27nffv 6341 . . . . . . . 8 𝑠(𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})
29 nfcv 2913 . . . . . . . . 9 𝑠((𝐹𝑛) “ 𝐴)
30 nfcv 2913 . . . . . . . . . 10 𝑠dom (𝐹𝑛)
3128, 30nfin 3969 . . . . . . . . 9 𝑠((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛))
3229, 31nfeq 2925 . . . . . . . 8 𝑠((𝐹𝑛) “ 𝐴) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛))
33 ineq1 3958 . . . . . . . . 9 (𝑠 = (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) → (𝑠 ∩ dom (𝐹𝑛)) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛)))
3433eqeq2d 2781 . . . . . . . 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 3511 . . . . . . 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 208 . . . . . 6 ((𝜑𝑛𝑍) → ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ 𝑆 ∧ ((𝐹𝑛) “ 𝐴) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛))))
3736simprd 483 . . . . 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 3366 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∈ V)
4038, 39fvmpt2d 6437 . . . . . 6 ((𝜑𝑛𝑍) → (𝐻𝑛) = (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
4140ineq1d 3964 . . . . 5 ((𝜑𝑛𝑍) → ((𝐻𝑛) ∩ dom (𝐹𝑛)) = ((𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}) ∩ dom (𝐹𝑛)))
4237, 41eqtr4d 2808 . . . 4 ((𝜑𝑛𝑍) → ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))
4342ex 397 . . 3 (𝜑 → (𝑛𝑍 → ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛))))
441, 43ralrimi 3106 . 2 (𝜑 → ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))
45 smfpimcclem.z . . . . . 6 𝑍𝑉
4645elexi 3365 . . . . 5 𝑍 ∈ V
4746mptex 6633 . . . 4 (𝑛𝑍 ↦ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) ∈ V
4824, 47eqeltri 2846 . . 3 𝐻 ∈ V
49 feq1 6165 . . . 4 ( = 𝐻 → (:𝑍𝑆𝐻:𝑍𝑆))
50 nfcv 2913 . . . . . 6 𝑛
51 nfmpt1 4882 . . . . . . 7 𝑛(𝑛𝑍 ↦ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))
5224, 51nfcxfr 2911 . . . . . 6 𝑛𝐻
5350, 52nfeq 2925 . . . . 5 𝑛 = 𝐻
54 fveq1 6332 . . . . . . 7 ( = 𝐻 → (𝑛) = (𝐻𝑛))
5554ineq1d 3964 . . . . . 6 ( = 𝐻 → ((𝑛) ∩ dom (𝐹𝑛)) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))
5655eqeq2d 2781 . . . . 5 ( = 𝐻 → (((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛)) ↔ ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛))))
5753, 56ralbid 3132 . . . 4 ( = 𝐻 → (∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛)) ↔ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛))))
5849, 57anbi12d 616 . . 3 ( = 𝐻 → ((:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))) ↔ (𝐻:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))))
5948, 58spcev 3451 . 2 ((𝐻:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝐻𝑛) ∩ dom (𝐹𝑛))) → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
6025, 44, 59syl2anc 573 1 (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wex 1852  wnf 1856  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  cin 3722  cmpt 4864  ccnv 5249  dom cdm 5250  ran crn 5251  cima 5253  wf 6026  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038
This theorem is referenced by:  smfpimcc  41531
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