Theorem bnj579 35166

Description: Technical lemma for bnj852 35173. 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 263	. 2 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		biid 263	. 2 ⊢ ([𝑔 / 𝑓]𝜑 ↔ [𝑔 / 𝑓]𝜑)
5		biid 263	. 2 ⊢ ([𝑔 / 𝑓]𝜓 ↔ [𝑔 / 𝑓]𝜓)
6		biid 263	. 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
7		bnj579.3	. 2 ⊢ 𝐷 = (ω ∖ {∅})
8		biid 263	. 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 263	. 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 35165	1 ⊢ (𝑛 ∈ 𝐷 → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∃*wmo 2558  ∀wral 3070  [wsbc 3739   ∖ cdif 3896  ∅c0 4280  {csn 4576  ∪ ciun 4943   class class class wbr 5094   E cep 5539  suc csuc 6337   Fn wfn 6505  ‘cfv 6510  ωcom 7835   predc-bnj14 34941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-fv 6518  df-om 7836  df-bnj17 34940

This theorem is referenced by:  bnj600  35171