Theorem bnj969 31016

Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj969.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj969.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj969.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj969.10	⊢ 𝐷 = (ω ∖ {∅})
bnj969.12	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj969.14	⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj969.15	⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛))

Assertion

Ref	Expression
bnj969	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)

Distinct variable groups:   𝐴,𝑖,𝑚,𝑦   𝑅,𝑖,𝑚,𝑦   𝑓,𝑖,𝑚,𝑦   𝑖,𝑛,𝑚

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜏(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑓,𝑛,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑓,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj969

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
2		bnj667 30822	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
3		bnj969.3	. . . . . . 7 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		bnj969.14	. . . . . . 7 ⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5	2, 3, 4	3imtr4i 281	. . . . . 6 ⊢ (𝜒 → 𝜏)
6	5	3ad2ant1 1082	. . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝜏)
7	6	adantl 482	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜏)
8	3	bnj1232 30874	. . . . . . 7 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
9		vex 3203	. . . . . . . 8 ⊢ 𝑚 ∈ V
10	9	bnj216 30800	. . . . . . 7 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛)
11		id 22	. . . . . . 7 ⊢ (𝑝 = suc 𝑛 → 𝑝 = suc 𝑛)
12	8, 10, 11	3anim123i 1247	. . . . . 6 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
13		bnj969.15	. . . . . . 7 ⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛))
14		3ancomb 1047	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛) ↔ (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
15	13, 14	bitri 264	. . . . . 6 ⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
16	12, 15	sylibr 224	. . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝜎)
17	16	adantl 482	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜎)
18	1, 7, 17	jca32 558	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜏 ∧ 𝜎)))
19		bnj256 30772	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜏 ∧ 𝜎)))
20	18, 19	sylibr 224	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎))
21		bnj969.12	. . 3 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
22		bnj969.10	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
23		bnj969.1	. . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
24		bnj969.2	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
25	22, 4, 13, 23, 24	bnj938 31007	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) ∈ V)
26	21, 25	syl5eqel 2705	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐶 ∈ V)
27	20, 26	syl 17	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  {csn 4177  ∪ ciun 4520  suc csuc 5725   Fn wfn 5883  ‘cfv 5888  ωcom 7065   ∧ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj910  31018  bnj1006  31029