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Theorem bnj1014 31844
 Description: Technical lemma for bnj69 31892. 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 2951 . . . . . . . . 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 31816 . . . . . . . . . 10 ((𝑓 Fn 𝑛𝜑𝜓) → ∀𝑖(𝑓 Fn 𝑛𝜑𝜓))
65nf5i 2119 . . . . . . . . 9 𝑖(𝑓 Fn 𝑛𝜑𝜓)
72, 6nfrex 3273 . . . . . . . 8 𝑖𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
87nfab 2957 . . . . . . 7 𝑖{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
91, 8nfcxfr 2949 . . . . . 6 𝑖𝐵
109nfcri 2945 . . . . 5 𝑖 𝑔𝐵
11 nfv 1896 . . . . 5 𝑖 𝑗 ∈ dom 𝑔
1210, 11nfan 1885 . . . 4 𝑖(𝑔𝐵𝑗 ∈ dom 𝑔)
13 nfv 1896 . . . 4 𝑖(𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)
1412, 13nfim 1882 . . 3 𝑖((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
1514nf5ri 2161 . 2 (((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) → ∀𝑖((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
16 eleq1w 2867 . . . . . 6 (𝑗 = 𝑖 → (𝑗 ∈ dom 𝑔𝑖 ∈ dom 𝑔))
1716anbi2d 628 . . . . 5 (𝑗 = 𝑖 → ((𝑔𝐵𝑗 ∈ dom 𝑔) ↔ (𝑔𝐵𝑖 ∈ dom 𝑔)))
18 fveq2 6545 . . . . . 6 (𝑗 = 𝑖 → (𝑔𝑗) = (𝑔𝑖))
1918sseq1d 3925 . . . . 5 (𝑗 = 𝑖 → ((𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
2017, 19imbi12d 346 . . . 4 (𝑗 = 𝑖 → (((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
2120equcoms 2008 . . 3 (𝑖 = 𝑗 → (((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
221bnj1317 31706 . . . . . . 7 (𝑔𝐵 → ∀𝑓 𝑔𝐵)
2322nf5i 2119 . . . . . 6 𝑓 𝑔𝐵
24 nfv 1896 . . . . . 6 𝑓 𝑖 ∈ dom 𝑔
2523, 24nfan 1885 . . . . 5 𝑓(𝑔𝐵𝑖 ∈ dom 𝑔)
26 nfv 1896 . . . . 5 𝑓(𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)
2725, 26nfim 1882 . . . 4 𝑓((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
28 eleq1w 2867 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝐵𝑔𝐵))
29 dmeq 5665 . . . . . . 7 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
3029eleq2d 2870 . . . . . 6 (𝑓 = 𝑔 → (𝑖 ∈ dom 𝑓𝑖 ∈ dom 𝑔))
3128, 30anbi12d 630 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝐵𝑖 ∈ dom 𝑓) ↔ (𝑔𝐵𝑖 ∈ dom 𝑔)))
32 fveq1 6544 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝑖) = (𝑔𝑖))
3332sseq1d 3925 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3431, 33imbi12d 346 . . . 4 (𝑓 = 𝑔 → (((𝑓𝐵𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
35 ssiun2 4876 . . . . 5 (𝑖 ∈ dom 𝑓 → (𝑓𝑖) ⊆ 𝑖 ∈ dom 𝑓(𝑓𝑖))
36 ssiun2 4876 . . . . . 6 (𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖))
37 bnj1014.13 . . . . . . 7 𝐷 = (ω ∖ {∅})
383, 4, 37, 1bnj882 31810 . . . . . 6 trCl(𝑋, 𝐴, 𝑅) = 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖)
3936, 38syl6sseqr 3945 . . . . 5 (𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
4035, 39sylan9ssr 3909 . . . 4 ((𝑓𝐵𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
4127, 34, 40chvar 2371 . . 3 ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
4221, 41spei 2370 . 2 𝑖((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
4315, 42bnj1131 31672 1 ((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083  {cab 2777  ∀wral 3107  ∃wrex 3108   ∖ cdif 3862   ⊆ wss 3865  ∅c0 4217  {csn 4478  ∪ ciun 4831  dom cdm 5450  suc csuc 6075   Fn wfn 6227  ‘cfv 6232  ωcom 7443   predc-bnj14 31571   trClc-bnj18 31577 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-dm 5460  df-iota 6196  df-fv 6240  df-bnj18 31578 This theorem is referenced by:  bnj1015  31845
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