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Theorem bnj543 31191
Description: Technical lemma for bnj852 31219. 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
bnj543.1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj543.2 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj543.3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj543.4 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj543.5 (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
Assertion
Ref Expression
bnj543 ((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)   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 bnj543
StepHypRef Expression
1 bnj257 31003 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚))
2 bnj268 31005 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
31, 2bitri 264 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
4 bnj253 31000 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
5 bnj256 31002 . . . . . 6 (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
63, 4, 53bitr3i 290 . . . . 5 ((((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
7 bnj256 31002 . . . . . 6 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
873anbi1i 1390 . . . . 5 (((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
9 bnj543.4 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
10 bnj170 30994 . . . . . . 7 ((𝑓 Fn 𝑚𝜑′𝜓′) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
119, 10bitri 264 . . . . . 6 (𝜏 ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
12 bnj543.5 . . . . . . 7 (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
13 3anan32 1083 . . . . . . 7 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚) ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1412, 13bitri 264 . . . . . 6 (𝜎 ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1511, 14anbi12i 735 . . . . 5 ((𝜏𝜎) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
166, 8, 153bitr4ri 293 . . . 4 ((𝜏𝜎) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
1716anbi2i 732 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜏𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚)))
18 3anass 1081 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏𝜎)))
19 bnj252 30999 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚)))
2017, 18, 193bitr4i 292 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
21 df-suc 5842 . . . . . . 7 suc 𝑚 = (𝑚 ∪ {𝑚})
2221eqeq2i 2736 . . . . . 6 (𝑛 = suc 𝑚𝑛 = (𝑚 ∪ {𝑚}))
23223anbi2i 1391 . . . . 5 (((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚))
2423anbi2i 732 . . . 4 ((𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚)))
25 bnj252 30999 . . . 4 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚)))
2624, 19, 253bitr4i 292 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚))
27 bnj543.1 . . . 4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
28 bnj543.2 . . . 4 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
29 bnj543.3 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
30 biid 251 . . . 4 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
3127, 28, 29, 30bnj535 31188 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3226, 31sylbi 207 . 2 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3320, 32sylbi 207 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  cun 3678  c0 4023  {csn 4285  cop 4291   ciun 4628  suc csuc 5838   Fn wfn 5996  cfv 6001  ωcom 7182  w-bnj17 30982   predc-bnj14 30984   FrSe w-bnj15 30988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066  ax-reg 8613
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-om 7183  df-bnj17 30983  df-bnj14 30985  df-bnj13 30987  df-bnj15 30989
This theorem is referenced by:  bnj544  31192
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