Theorem bnj579 35410

Description: Technical lemma for bnj852 35417. 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
bnj579.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj579.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj579.3	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj579	⊢ (𝑛 ∈ 𝐷 → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Distinct variable groups:   𝐴,𝑓,𝑖   𝐷,𝑓   𝑅,𝑓,𝑖   𝑓,𝑛,𝑖   𝑥,𝑓   𝑦,𝑓,𝑖

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑓, 𝑖, 𝑛)   𝜓(𝑥, 𝑦, 𝑓, 𝑖, 𝑛)   𝐴(𝑥, 𝑦, 𝑛)   𝐷(𝑥, 𝑦, 𝑖, 𝑛)   𝑅(𝑥, 𝑦, 𝑛)

Proof of Theorem bnj579

Dummy variables 𝑘 𝑔 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj579.1	. 2 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2		bnj579.2	. 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		biid 264	. 2 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		biid 264	. 2 ⊢ ([𝑔 / 𝑓]𝜑 ↔ [𝑔 / 𝑓]𝜑)
5		biid 264	. 2 ⊢ ([𝑔 / 𝑓]𝜓 ↔ [𝑔 / 𝑓]𝜓)
6		biid 264	. 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
7		bnj579.3	. 2 ⊢ 𝐷 = (ω ∖ {∅})
8		biid 264	. 2 ⊢ (((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) → (𝑓‘𝑗) = (𝑔‘𝑗)))
9		biid 264	. 2 ⊢ (∀𝑘 ∈ 𝑛 (𝑘 E 𝑗 → [𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) → (𝑓‘𝑗) = (𝑔‘𝑗))) ↔ ∀𝑘 ∈ 𝑛 (𝑘 E 𝑗 → [𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) → (𝑓‘𝑗) = (𝑔‘𝑗))))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	bnj580 35409	1 ⊢ (𝑛 ∈ 𝐷 → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃*wmo 2564  ∀wral 3078  [wsbc 3742   ∖ cdif 3899  ∅c0 4282  {csn 4587  ∪ ciun 4954   class class class wbr 5107   E cep 5558  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7865   predc-bnj14 35185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-om 7866  df-bnj17 35184

This theorem is used by:  bnj600  35415