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Theorem bnj546 31304
Description: Technical lemma for bnj852 31329. 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
bnj546.1 𝐷 = (ω ∖ {∅})
bnj546.2 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj546.3 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj546.4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj546.5 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj546 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
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)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj546
StepHypRef Expression
1 bnj546.2 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
2 3simpc 1146 . . . . . . 7 ((𝑓 Fn 𝑚𝜑′𝜓′) → (𝜑′𝜓′))
31, 2sylbi 207 . . . . . 6 (𝜏 → (𝜑′𝜓′))
4 bnj546.3 . . . . . . 7 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
5 bnj546.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
65bnj923 31176 . . . . . . . . 9 (𝑚𝐷𝑚 ∈ ω)
763ad2ant1 1127 . . . . . . . 8 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → 𝑚 ∈ ω)
8 simp3 1132 . . . . . . . 8 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → 𝑝𝑚)
97, 8jca 501 . . . . . . 7 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → (𝑚 ∈ ω ∧ 𝑝𝑚))
104, 9sylbi 207 . . . . . 6 (𝜎 → (𝑚 ∈ ω ∧ 𝑝𝑚))
113, 10anim12i 600 . . . . 5 ((𝜏𝜎) → ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
12 bnj256 31112 . . . . 5 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
1311, 12sylibr 224 . . . 4 ((𝜏𝜎) → (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
1413anim2i 603 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜏𝜎)) → (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)))
15143impb 1107 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)))
16 bnj546.4 . . 3 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
17 bnj546.5 . . 3 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
18 biid 251 . . 3 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
1916, 17, 18bnj518 31294 . 2 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
20 fvex 6342 . . 3 (𝑓𝑝) ∈ V
21 iunexg 7290 . . 3 (((𝑓𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
2220, 21mpan 670 . 2 (∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
2315, 19, 223syl 18 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cdif 3720  c0 4063  {csn 4316   ciun 4654  suc csuc 5868   Fn wfn 6026  cfv 6031  ωcom 7212  w-bnj17 31092   predc-bnj14 31094   FrSe w-bnj15 31098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-bnj17 31093  df-bnj14 31095  df-bnj13 31097  df-bnj15 31099
This theorem is referenced by:  bnj938  31345
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