Theorem bnj579 30984

Description: Technical lemma for bnj852 30991. 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 251	. 2 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		biid 251	. 2 ⊢ ([𝑔 / 𝑓]𝜑 ↔ [𝑔 / 𝑓]𝜑)
5		biid 251	. 2 ⊢ ([𝑔 / 𝑓]𝜓 ↔ [𝑔 / 𝑓]𝜓)
6		biid 251	. 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
7		bnj579.3	. 2 ⊢ 𝐷 = (ω ∖ {∅})
8		biid 251	. 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 251	. 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 30983	1 ⊢ (𝑛 ∈ 𝐷 → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃*wmo 2471  ∀wral 2912  [wsbc 3435   ∖ cdif 3571  ∅c0 3915  {csn 4177  ∪ ciun 4520   class class class wbr 4653   E cep 5028  suc csuc 5725   Fn wfn 5883  ‘cfv 5888  ωcom 7065   predc-bnj14 30754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj600  30989