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Theorem bnj544 31567
 Description: Technical lemma for bnj852 31594. 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
bnj544.1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj544.2 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj544.3 𝐷 = (ω ∖ {∅})
bnj544.4 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj544.5 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj544.6 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
Assertion
Ref Expression
bnj544 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′
Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑚,𝑛)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑥,𝑓,𝑚,𝑛)   𝐺(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj544
StepHypRef Expression
1 bnj544.6 . . 3 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
2 bnj544.3 . . . . 5 𝐷 = (ω ∖ {∅})
32bnj923 31441 . . . 4 (𝑚𝐷𝑚 ∈ ω)
433anim1i 1152 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
51, 4sylbi 209 . 2 (𝜎 → (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
6 bnj544.1 . . 3 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
7 bnj544.2 . . 3 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
8 bnj544.4 . . 3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
9 bnj544.5 . . 3 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
10 biid 253 . . 3 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚) ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
116, 7, 8, 9, 10bnj543 31566 . 2 ((𝑅 FrSe 𝐴𝜏 ∧ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚)) → 𝐺 Fn 𝑛)
125, 11syl3an3 1166 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090   ∖ cdif 3789   ∪ cun 3790  ∅c0 4141  {csn 4398  ⟨cop 4404  ∪ ciun 4755  suc csuc 5980   Fn wfn 6132  ‘cfv 6137  ωcom 7345   predc-bnj14 31360   FrSe w-bnj15 31364 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228  ax-reg 8788 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-om 7346  df-bnj17 31359  df-bnj14 31361  df-bnj13 31363  df-bnj15 31365 This theorem is referenced by:  bnj600  31592  bnj908  31604
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