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Theorem bnj1014 30284
Description: Technical lemma for bnj69 30332. 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 2751 . . . . . . . . 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 30256 . . . . . . . . . 10 ((𝑓 Fn 𝑛𝜑𝜓) → ∀𝑖(𝑓 Fn 𝑛𝜑𝜓))
65nf5i 2011 . . . . . . . . 9 𝑖(𝑓 Fn 𝑛𝜑𝜓)
72, 6nfrex 2990 . . . . . . . 8 𝑖𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
87nfab 2755 . . . . . . 7 𝑖{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
91, 8nfcxfr 2749 . . . . . 6 𝑖𝐵
109nfcri 2745 . . . . 5 𝑖 𝑔𝐵
11 nfv 1830 . . . . 5 𝑖 𝑗 ∈ dom 𝑔
1210, 11nfan 1816 . . . 4 𝑖(𝑔𝐵𝑗 ∈ dom 𝑔)
13 nfv 1830 . . . 4 𝑖(𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)
1412, 13nfim 1813 . . 3 𝑖((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
1514nf5ri 2053 . 2 (((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) → ∀𝑖((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
16 eleq1 2676 . . . . . 6 (𝑗 = 𝑖 → (𝑗 ∈ dom 𝑔𝑖 ∈ dom 𝑔))
1716anbi2d 736 . . . . 5 (𝑗 = 𝑖 → ((𝑔𝐵𝑗 ∈ dom 𝑔) ↔ (𝑔𝐵𝑖 ∈ dom 𝑔)))
18 fveq2 6103 . . . . . 6 (𝑗 = 𝑖 → (𝑔𝑗) = (𝑔𝑖))
1918sseq1d 3595 . . . . 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 1934 . . 3 (𝑖 = 𝑗 → (((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
221bnj1317 30146 . . . . . . 7 (𝑔𝐵 → ∀𝑓 𝑔𝐵)
2322nf5i 2011 . . . . . 6 𝑓 𝑔𝐵
24 nfv 1830 . . . . . 6 𝑓 𝑖 ∈ dom 𝑔
2523, 24nfan 1816 . . . . 5 𝑓(𝑔𝐵𝑖 ∈ dom 𝑔)
26 nfv 1830 . . . . 5 𝑓(𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)
2725, 26nfim 1813 . . . 4 𝑓((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
28 eleq1 2676 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝐵𝑔𝐵))
29 dmeq 5246 . . . . . . 7 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
3029eleq2d 2673 . . . . . 6 (𝑓 = 𝑔 → (𝑖 ∈ dom 𝑓𝑖 ∈ dom 𝑔))
3128, 30anbi12d 743 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝐵𝑖 ∈ dom 𝑓) ↔ (𝑔𝐵𝑖 ∈ dom 𝑔)))
32 fveq1 6102 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝑖) = (𝑔𝑖))
3332sseq1d 3595 . . . . 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 4499 . . . . 5 (𝑖 ∈ dom 𝑓 → (𝑓𝑖) ⊆ 𝑖 ∈ dom 𝑓(𝑓𝑖))
36 ssiun2 4499 . . . . . 6 (𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖))
37 bnj1014.13 . . . . . . 7 𝐷 = (ω ∖ {∅})
383, 4, 37, 1bnj882 30250 . . . . . 6 trCl(𝑋, 𝐴, 𝑅) = 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖)
3936, 38syl6sseqr 3615 . . . . 5 (𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
4035, 39sylan9ssr 3582 . . . 4 ((𝑓𝐵𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
4127, 34, 40chvar 2250 . . 3 ((𝑔𝐵𝑖 ∈ dom 𝑔) → (𝑔𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
4221, 41spei 2249 . 2 𝑖((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
4315, 42bnj1131 30112 1 ((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  {cab 2596  wral 2896  wrex 2897  cdif 3537  wss 3540  c0 3874  {csn 4125   ciun 4455  dom cdm 5038  suc csuc 5642   Fn wfn 5799  cfv 5804  ωcom 6957   predc-bnj14 30007   trClc-bnj18 30013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-bnj18 30014
This theorem is referenced by:  bnj1015  30285
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