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Theorem findcard2d 8748
Description: Deduction version of findcard2 8746. (Contributed by SO, 16-Jul-2018.)
Hypotheses
Ref Expression
findcard2d.ch (𝑥 = ∅ → (𝜓𝜒))
findcard2d.th (𝑥 = 𝑦 → (𝜓𝜃))
findcard2d.ta (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓𝜏))
findcard2d.et (𝑥 = 𝐴 → (𝜓𝜂))
findcard2d.z (𝜑𝜒)
findcard2d.i ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (𝜃𝜏))
findcard2d.a (𝜑𝐴 ∈ Fin)
Assertion
Ref Expression
findcard2d (𝜑𝜂)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝜓,𝑦,𝑧   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜂,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦,𝑧)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧)   𝜂(𝑦,𝑧)

Proof of Theorem findcard2d
StepHypRef Expression
1 ssid 3986 . 2 𝐴𝐴
2 findcard2d.a . . . 4 (𝜑𝐴 ∈ Fin)
32adantr 481 . . 3 ((𝜑𝐴𝐴) → 𝐴 ∈ Fin)
4 sseq1 3989 . . . . . 6 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
54anbi2d 628 . . . . 5 (𝑥 = ∅ → ((𝜑𝑥𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
6 findcard2d.ch . . . . 5 (𝑥 = ∅ → (𝜓𝜒))
75, 6imbi12d 346 . . . 4 (𝑥 = ∅ → (((𝜑𝑥𝐴) → 𝜓) ↔ ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)))
8 sseq1 3989 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98anbi2d 628 . . . . 5 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
10 findcard2d.th . . . . 5 (𝑥 = 𝑦 → (𝜓𝜃))
119, 10imbi12d 346 . . . 4 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝜓) ↔ ((𝜑𝑦𝐴) → 𝜃)))
12 sseq1 3989 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
1312anbi2d 628 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑𝑥𝐴) ↔ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)))
14 findcard2d.ta . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓𝜏))
1513, 14imbi12d 346 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝜑𝑥𝐴) → 𝜓) ↔ ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
16 sseq1 3989 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
1716anbi2d 628 . . . . 5 (𝑥 = 𝐴 → ((𝜑𝑥𝐴) ↔ (𝜑𝐴𝐴)))
18 findcard2d.et . . . . 5 (𝑥 = 𝐴 → (𝜓𝜂))
1917, 18imbi12d 346 . . . 4 (𝑥 = 𝐴 → (((𝜑𝑥𝐴) → 𝜓) ↔ ((𝜑𝐴𝐴) → 𝜂)))
20 findcard2d.z . . . . 5 (𝜑𝜒)
2120adantr 481 . . . 4 ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)
22 simprl 767 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝜑)
23 simprr 769 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
2423unssad 4160 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑦𝐴)
2522, 24jca 512 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜑𝑦𝐴))
26 id 22 . . . . . . . . . . 11 ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
27 vsnid 4592 . . . . . . . . . . . 12 𝑧 ∈ {𝑧}
28 elun2 4150 . . . . . . . . . . . 12 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
2927, 28mp1i 13 . . . . . . . . . . 11 ((𝑦 ∪ {𝑧}) ⊆ 𝐴𝑧 ∈ (𝑦 ∪ {𝑧}))
3026, 29sseldd 3965 . . . . . . . . . 10 ((𝑦 ∪ {𝑧}) ⊆ 𝐴𝑧𝐴)
3130ad2antll 725 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧𝐴)
32 simplr 765 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧𝑦)
3331, 32eldifd 3944 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ (𝐴𝑦))
34 findcard2d.i . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (𝜃𝜏))
3522, 24, 33, 34syl12anc 832 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜃𝜏))
3625, 35embantd 59 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (((𝜑𝑦𝐴) → 𝜃) → 𝜏))
3736ex 413 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (((𝜑𝑦𝐴) → 𝜃) → 𝜏)))
3837com23 86 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((𝜑𝑦𝐴) → 𝜃) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
397, 11, 15, 19, 21, 38findcard2s 8747 . . 3 (𝐴 ∈ Fin → ((𝜑𝐴𝐴) → 𝜂))
403, 39mpcom 38 . 2 ((𝜑𝐴𝐴) → 𝜂)
411, 40mpan2 687 1 (𝜑𝜂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  cdif 3930  cun 3931  wss 3933  c0 4288  {csn 4557  Fincfn 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1o 8091  df-er 8278  df-en 8498  df-fin 8501
This theorem is referenced by:  fprodmodd  15339  sumeven  15726  sumodd  15727  maducoeval2  21177  madugsum  21180  esum2dlem  31250  fiunelcarsg  31473  carsgclctunlem1  31474  fiiuncl  41204  mpct  41340  fprodexp  41751  fprodabs2  41752  mccl  41755  fprodcn  41757  fprodcncf  42060  dvnprodlem3  42109  sge0iunmptlemfi  42572  hoidmvle  42759
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