Theorem findcard2d 9085

Description: Deduction version of findcard2 9083. (Contributed by SO, 16-Jul-2018.)

Hypotheses

Ref	Expression
findcard2d.ch	⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒))
findcard2d.th	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))
findcard2d.ta	⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))
findcard2d.et	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))
findcard2d.z	⊢ (𝜑 → 𝜒)
findcard2d.i	⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))
findcard2d.a	⊢ (𝜑 → 𝐴 ∈ Fin)

Assertion

Ref	Expression
findcard2d	⊢ (𝜑 → 𝜂)

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

Step	Hyp	Ref	Expression
1		ssid 3953	. 2 ⊢ 𝐴 ⊆ 𝐴
2		findcard2d.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
3	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝐴 ∈ Fin)
4		sseq1 3956	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
5	4	anbi2d 630	. . . . 5 ⊢ (𝑥 = ∅ → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
6		findcard2d.ch	. . . . 5 ⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒))
7	5, 6	imbi12d 344	. . . 4 ⊢ (𝑥 = ∅ → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)))
8		sseq1 3956	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
9	8	anbi2d 630	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ 𝑦 ⊆ 𝐴)))
10		findcard2d.th	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))
11	9, 10	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃)))
12		sseq1 3956	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
13	12	anbi2d 630	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)))
14		findcard2d.ta	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))
15	13, 14	imbi12d 344	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
16		sseq1 3956	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
17	16	anbi2d 630	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐴)))
18		findcard2d.et	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))
19	17, 18	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝜂)))
20		findcard2d.z	. . . . 5 ⊢ (𝜑 → 𝜒)
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)
22		simprl 770	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝜑)
23		simprr 772	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
24	23	unssad 4142	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑦 ⊆ 𝐴)
25	22, 24	jca 511	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜑 ∧ 𝑦 ⊆ 𝐴))
26		id 22	. . . . . . . . . . 11 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
27		vsnid 4617	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ {𝑧}
28		elun2 4132	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
29	27, 28	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑧 ∈ (𝑦 ∪ {𝑧}))
30	26, 29	sseldd 3931	. . . . . . . . . 10 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑧 ∈ 𝐴)
31	30	ad2antll 729	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ 𝐴)
32		simplr 768	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧 ∈ 𝑦)
33	31, 32	eldifd 3909	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ (𝐴 ∖ 𝑦))
34		findcard2d.i	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))
35	22, 24, 33, 34	syl12anc 836	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜃 → 𝜏))
36	25, 35	embantd 59	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃) → 𝜏))
37	36	ex 412	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃) → 𝜏)))
38	37	com23 86	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
39	7, 11, 15, 19, 21, 38	findcard2s 9084	. . 3 ⊢ (𝐴 ∈ Fin → ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝜂))
40	3, 39	mpcom 38	. 2 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝜂)
41	1, 40	mpan2 691	1 ⊢ (𝜑 → 𝜂)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3895   ∪ cun 3896   ⊆ wss 3898  ∅c0 4282  {csn 4577  Fincfn 8877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-om 7805  df-en 8878  df-fin 8881

This theorem is referenced by:  fprodmodd  15908  sumeven  16302  sumodd  16303  maducoeval2  22558  madugsum  22561  elrgspnlem4  33221  rprmdvdsprod  33508  constrextdg2lem  33784  constrfiss  33787  esum2dlem  34128  fiunelcarsg  34352  carsgclctunlem1  34353  evl1gprodd  42233  idomnnzgmulnz  42249  deg1gprod  42256  fiiuncl  45189  mpct  45325  fprodexp  45721  fprodabs2  45722  mccl  45725  fprodcn  45727  fprodcncf  46025  dvnprodlem3  46073  sge0iunmptlemfi  46538  hoidmvle  46725