Theorem bnj938 35295

Description: Technical lemma for bnj69 35368. 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
bnj938.1	⊢ 𝐷 = (ω ∖ {∅})
bnj938.2	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj938.3	⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj938.4	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj938.5	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj938	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 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

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

Proof of Theorem bnj938

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2852	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 = 𝑋)
2	1	bnj706 35113	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∃𝑥 𝑥 = 𝑋)
3		bnj291 35070	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑋 ∈ 𝐴))
4	3	simplbi 501	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎))
5		bnj602 35273	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
6	5	eqeq2d 2781	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
7		bnj938.4	. . . . . . . . 9 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
8	6, 7	bitr4di 292	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ 𝜑′))
9	8	3anbi2d 1467	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)))
10		bnj938.2	. . . . . . 7 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
11	9, 10	bitr4di 292	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ 𝜏))
12	11	3anbi2d 1467	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎)))
13	4, 12	imbitrrid 249	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎)))
14		bnj938.1	. . . . 5 ⊢ 𝐷 = (ω ∖ {∅})
15		biid 264	. . . . 5 ⊢ ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′))
16		bnj938.3	. . . . 5 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
17		biid 264	. . . . 5 ⊢ ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
18		bnj938.5	. . . . 5 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
19	14, 15, 16, 17, 18	bnj546 35254	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
20	13, 19	syl6 36	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V))
21	20	exlimiv 1958	. 2 ⊢ (∃𝑥 𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V))
22	2, 21	mpcom 39	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101   = wceq 1568  ∃wex 1807   ∈ wcel 2150  ∀wral 3086  Vcvv 3462   ∖ cdif 3910  ∅c0 4294  {csn 4594  ∪ ciun 4961  suc csuc 6366   Fn wfn 6535  ‘cfv 6540  ωcom 7865   ∧ w-bnj17 35045   predc-bnj14 35047   FrSe w-bnj15 35051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-mo 2574  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fv 6548  df-om 7866  df-bnj17 35046  df-bnj14 35048  df-bnj13 35050  df-bnj15 35052

This theorem is referenced by:  bnj944  35296  bnj969  35304