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Theorem bnj969 31615
 Description: Technical lemma for bnj69 31677. 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
bnj969.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj969.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj969.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj969.10 𝐷 = (ω ∖ {∅})
bnj969.12 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj969.14 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
bnj969.15 (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
Assertion
Ref Expression
bnj969 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
Distinct variable groups:   𝐴,𝑖,𝑚,𝑦   𝑅,𝑖,𝑚,𝑦   𝑓,𝑖,𝑚,𝑦   𝑖,𝑛,𝑚
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜏(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑓,𝑛,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑓,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj969
StepHypRef Expression
1 simpl 476 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴𝑋𝐴))
2 bnj667 31421 . . . . . . 7 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → (𝑓 Fn 𝑛𝜑𝜓))
3 bnj969.3 . . . . . . 7 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj969.14 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
52, 3, 43imtr4i 284 . . . . . 6 (𝜒𝜏)
653ad2ant1 1124 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜏)
76adantl 475 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜏)
83bnj1232 31473 . . . . . . 7 (𝜒𝑛𝐷)
9 vex 3401 . . . . . . . 8 𝑚 ∈ V
109bnj216 31400 . . . . . . 7 (𝑛 = suc 𝑚𝑚𝑛)
11 id 22 . . . . . . 7 (𝑝 = suc 𝑛𝑝 = suc 𝑛)
128, 10, 113anim123i 1151 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
13 bnj969.15 . . . . . . 7 (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
14 3ancomb 1084 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
1513, 14bitri 267 . . . . . 6 (𝜎 ↔ (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
1612, 15sylibr 226 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜎)
1716adantl 475 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜎)
181, 7, 17jca32 511 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜏𝜎)))
19 bnj256 31374 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜏𝜎)))
2018, 19sylibr 226 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎))
21 bnj969.12 . . 3 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
22 bnj969.10 . . . 4 𝐷 = (ω ∖ {∅})
23 bnj969.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
24 bnj969.2 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2522, 4, 13, 23, 24bnj938 31606 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅) ∈ V)
2621, 25syl5eqel 2863 . 2 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝐶 ∈ V)
2720, 26syl 17 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  Vcvv 3398   ∖ cdif 3789  ∅c0 4141  {csn 4398  ∪ ciun 4753  suc csuc 5978   Fn wfn 6130  ‘cfv 6135  ωcom 7343   ∧ w-bnj17 31354   predc-bnj14 31356   FrSe w-bnj15 31360 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 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226 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 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-bnj17 31355  df-bnj14 31357  df-bnj13 31359  df-bnj15 31361 This theorem is referenced by:  bnj910  31617  bnj1006  31628
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