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Theorem bnj1014 31156
Description: Technical lemma for bnj69 31204. 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
bnj1014.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1014.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1014.13 𝐷 = (ω ∖ {∅})
bnj1014.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj1014 ((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝑓,𝑔,𝑖   𝑖,𝑗   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑔,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝐴(𝑔,𝑗)   𝐵(𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑔,𝑗,𝑛)   𝑅(𝑔,𝑗)   𝑋(𝑔,𝑗)

Proof of Theorem bnj1014
StepHypRef Expression
1 bnj1014.14 . . . . . . 7 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
2 nfcv 2793 . . . . . . . . 9 𝑖𝐷
3 bnj1014.1 . . . . . . . . . . 11 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
4 bnj1014.2 . . . . . . . . . . 11 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
53, 4bnj911 31128 . . . . . . . . . 10 ((𝑓 Fn 𝑛𝜑𝜓) → ∀𝑖(𝑓 Fn 𝑛𝜑𝜓))
65nf5i 2064 . . . . . . . . 9 𝑖(𝑓 Fn 𝑛𝜑𝜓)
72, 6nfrex 3036 . . . . . . . 8 𝑖𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
87nfab 2798 . . . . . . 7 𝑖{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
91, 8nfcxfr 2791 . . . . . 6 𝑖𝐵
109nfcri 2787 . . . . 5 𝑖 𝑔𝐵
11 nfv 1883 . . . . 5 𝑖 𝑗 ∈ dom 𝑔
1210, 11nfan 1868 . . . 4 𝑖(𝑔𝐵𝑗 ∈ dom 𝑔)
13 nfv 1883 . . . 4 𝑖(𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)
1412, 13nfim 1865 . . 3 𝑖((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
1514nf5ri 2103 . 2 (((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) → ∀𝑖((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
16 eleq1 2718 . . . . . 6 (𝑗 = 𝑖 → (𝑗 ∈ dom 𝑔𝑖 ∈ dom 𝑔))
1716anbi2d 740 . . . . 5 (𝑗 = 𝑖 → ((𝑔𝐵𝑗 ∈ dom 𝑔) ↔ (𝑔𝐵𝑖 ∈ dom 𝑔)))
18 fveq2 6229 . . . . . 6 (𝑗 = 𝑖 → (𝑔𝑗) = (𝑔𝑖))
1918sseq1d 3665 . . . . 5 (𝑗 = 𝑖 → ((𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
2017, 19imbi12d 333 . . . 4 (𝑗 = 𝑖 → (((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
2120equcoms 1993 . . 3 (𝑖 = 𝑗 → (((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
221bnj1317 31018 . . . . . . 7 (𝑔𝐵 → ∀𝑓 𝑔𝐵)
2322nf5i 2064 . . . . . 6 𝑓 𝑔𝐵
24 nfv 1883 . . . . . 6 𝑓 𝑖 ∈ dom 𝑔
2523, 24nfan 1868 . . . . 5 𝑓(𝑔𝐵𝑖 ∈ dom 𝑔)
26 nfv 1883 . . . . 5 𝑓(𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)
2725, 26nfim 1865 . . . 4 𝑓((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
28 eleq1 2718 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝐵𝑔𝐵))
29 dmeq 5356 . . . . . . 7 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
3029eleq2d 2716 . . . . . 6 (𝑓 = 𝑔 → (𝑖 ∈ dom 𝑓𝑖 ∈ dom 𝑔))
3128, 30anbi12d 747 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝐵𝑖 ∈ dom 𝑓) ↔ (𝑔𝐵𝑖 ∈ dom 𝑔)))
32 fveq1 6228 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝑖) = (𝑔𝑖))
3332sseq1d 3665 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3431, 33imbi12d 333 . . . 4 (𝑓 = 𝑔 → (((𝑓𝐵𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
35 ssiun2 4595 . . . . 5 (𝑖 ∈ dom 𝑓 → (𝑓𝑖) ⊆ 𝑖 ∈ dom 𝑓(𝑓𝑖))
36 ssiun2 4595 . . . . . 6 (𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖))
37 bnj1014.13 . . . . . . 7 𝐷 = (ω ∖ {∅})
383, 4, 37, 1bnj882 31122 . . . . . 6 trCl(𝑋, 𝐴, 𝑅) = 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖)
3936, 38syl6sseqr 3685 . . . . 5 (𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
4035, 39sylan9ssr 3650 . . . 4 ((𝑓𝐵𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
4127, 34, 40chvar 2298 . . 3 ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
4221, 41spei 2297 . 2 𝑖((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
4315, 42bnj1131 30984 1 ((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  cdif 3604  wss 3607  c0 3948  {csn 4210   ciun 4552  dom cdm 5143  suc csuc 5763   Fn wfn 5921  cfv 5926  ωcom 7107   predc-bnj14 30882   trClc-bnj18 30888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-dm 5153  df-iota 5889  df-fv 5934  df-bnj18 30889
This theorem is referenced by:  bnj1015  31157
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