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Theorem bnj544 32193
Description: Technical lemma for bnj852 32220. 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 32066 . . . 4 (𝑚𝐷𝑚 ∈ ω)
433anim1i 1149 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
51, 4sylbi 220 . 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 264 . . 3 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚) ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
116, 7, 8, 9, 10bnj543 32192 . 2 ((𝑅 FrSe 𝐴𝜏 ∧ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚)) → 𝐺 Fn 𝑛)
125, 11syl3an3 1162 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2115  wral 3133  cdif 3916  cun 3917  c0 4276  {csn 4550  cop 4556   ciun 4906  suc csuc 6181   Fn wfn 6339  cfv 6344  ωcom 7571   predc-bnj14 31985   FrSe w-bnj15 31989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7452  ax-reg 9049
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-om 7572  df-bnj17 31984  df-bnj14 31986  df-bnj13 31988  df-bnj15 31990
This theorem is referenced by:  bnj600  32218  bnj908  32230
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