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Theorem bnj579 32190
 Description: Technical lemma for bnj852 32197. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj579.3 𝐷 = (ω ∖ {∅})
Assertion
Ref Expression
bnj579 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
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.
StepHypRef Expression
1 bnj579.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2 bnj579.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 biid 263 . 2 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
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 𝑛𝜑𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓))
7 bnj579.3 . 2 𝐷 = (ω ∖ {∅})
8 biid 263 . 2 (((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗)))
9 biid 263 . 2 (∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗))) ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗))))
101, 2, 3, 4, 5, 6, 7, 8, 9bnj580 32189 1 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃*wmo 2620  ∀wral 3125  [wsbc 3748   ∖ cdif 3906  ∅c0 4265  {csn 4539  ∪ ciun 4891   class class class wbr 5038   E cep 5436  suc csuc 6165   Fn wfn 6322  ‘cfv 6327  ωcom 7554   predc-bnj14 31962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-tr 5145  df-id 5432  df-eprel 5437  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6286  df-fun 6329  df-fn 6330  df-fv 6335  df-om 7555  df-bnj17 31961 This theorem is referenced by:  bnj600  32195
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