Theorem bnj546 32170

Description: Technical lemma for bnj852 32195. 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
bnj546.1	⊢ 𝐷 = (ω ∖ {∅})
bnj546.2	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj546.3	⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj546.4	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj546.5	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj546	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′

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

Proof of Theorem bnj546

Step	Hyp	Ref	Expression
1		bnj546.2	. . . . . . 7 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
2		3simpc 1146	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) → (𝜑′ ∧ 𝜓′))
3	1, 2	sylbi 219	. . . . . 6 ⊢ (𝜏 → (𝜑′ ∧ 𝜓′))
4		bnj546.3	. . . . . . 7 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
5		bnj546.1	. . . . . . . . . 10 ⊢ 𝐷 = (ω ∖ {∅})
6	5	bnj923 32041	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝐷 → 𝑚 ∈ ω)
7	6	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → 𝑚 ∈ ω)
8		simp3 1134	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → 𝑝 ∈ 𝑚)
9	7, 8	jca 514	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
10	4, 9	sylbi 219	. . . . . 6 ⊢ (𝜎 → (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
11	3, 10	anim12i 614	. . . . 5 ⊢ ((𝜏 ∧ 𝜎) → ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
12		bnj256 31978	. . . . 5 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
13	11, 12	sylibr 236	. . . 4 ⊢ ((𝜏 ∧ 𝜎) → (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
14	13	anim2i 618	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)) → (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
15	14	3impb 1111	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
16		bnj546.4	. . 3 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
17		bnj546.5	. . 3 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
18		biid 263	. . 3 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
19	16, 17, 18	bnj518 32160	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
20		fvex 6685	. . 3 ⊢ (𝑓‘𝑝) ∈ V
21		iunexg 7666	. . 3 ⊢ (((𝑓‘𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
22	20, 21	mpan 688	. 2 ⊢ (∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
23	15, 19, 22	3syl 18	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ∖ cdif 3935  ∅c0 4293  {csn 4569  ∪ ciun 4921  suc csuc 6195   Fn wfn 6352  ‘cfv 6357  ωcom 7582   ∧ w-bnj17 31958   predc-bnj14 31960   FrSe w-bnj15 31964

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-bnj17 31959  df-bnj14 31961  df-bnj13 31963  df-bnj15 31965

This theorem is referenced by:  bnj938  32211