Theorem bnj938 35095

Description: Technical lemma for bnj69 35168. 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 2819	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 = 𝑋)
2	1	bnj706 34912	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∃𝑥 𝑥 = 𝑋)
3		bnj291 34869	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑋 ∈ 𝐴))
4	3	simplbi 497	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎))
5		bnj602 35073	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
6	5	eqeq2d 2748	. . . . . . . . 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 289	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ 𝜑′))
9	8	3anbi2d 1444	. . . . . . 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 289	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ 𝜏))
12	11	3anbi2d 1444	. . . . 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 246	. . . 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 261	. . . . 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 261	. . . . 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 35054	. . . 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 35	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V))
21	20	exlimiv 1932	. 2 ⊢ (∃𝑥 𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V))
22	2, 21	mpcom 38	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3441   ∖ cdif 3899  ∅c0 4286  {csn 4581  ∪ ciun 4947  suc csuc 6320   Fn wfn 6488  ‘cfv 6493  ωcom 7810   ∧ w-bnj17 34844   predc-bnj14 34846   FrSe w-bnj15 34850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fv 6501  df-om 7811  df-bnj17 34845  df-bnj14 34847  df-bnj13 34849  df-bnj15 34851

This theorem is referenced by:  bnj944  35096  bnj969  35104