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Theorem bnj906 33542
Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj906 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))

Proof of Theorem bnj906
Dummy variables 𝑓 𝑖 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1onn 8586 . . . . . . . 8 1o ∈ ω
2 1n0 8434 . . . . . . . 8 1o ≠ ∅
3 eldifsn 4747 . . . . . . . 8 (1o ∈ (ω ∖ {∅}) ↔ (1o ∈ ω ∧ 1o ≠ ∅))
41, 2, 3mpbir2an 709 . . . . . . 7 1o ∈ (ω ∖ {∅})
54ne0ii 4297 . . . . . 6 (ω ∖ {∅}) ≠ ∅
6 biid 260 . . . . . . 7 ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
7 biid 260 . . . . . . 7 (∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
8 eqid 2736 . . . . . . 7 (ω ∖ {∅}) = (ω ∖ {∅})
96, 7, 8bnj852 33533 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛 ∈ (ω ∖ {∅})∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
10 r19.2z 4452 . . . . . 6 (((ω ∖ {∅}) ≠ ∅ ∧ ∀𝑛 ∈ (ω ∖ {∅})∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))) → ∃𝑛 ∈ (ω ∖ {∅})∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
115, 9, 10sylancr 587 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑛 ∈ (ω ∖ {∅})∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
12 euex 2575 . . . . 5 (∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → ∃𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
1311, 12bnj31 33331 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑛 ∈ (ω ∖ {∅})∃𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
14 rexcom4 3271 . . . 4 (∃𝑛 ∈ (ω ∖ {∅})∃𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∃𝑓𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
1513, 14sylib 217 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑓𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
16 abid 2717 . . 3 (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} ↔ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
1715, 16bnj1198 33407 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑓 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))})
18 simp2 1137 . . . . . . 7 ((𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
1918reximi 3087 . . . . . 6 (∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → ∃𝑛 ∈ (ω ∖ {∅})(𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2016, 19sylbi 216 . . . . 5 (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} → ∃𝑛 ∈ (ω ∖ {∅})(𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
21 df-rex 3074 . . . . . 6 (∃𝑛 ∈ (ω ∖ {∅})(𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ ∃𝑛(𝑛 ∈ (ω ∖ {∅}) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
22 19.41v 1953 . . . . . . 7 (∃𝑛(𝑛 ∈ (ω ∖ {∅}) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ↔ (∃𝑛 𝑛 ∈ (ω ∖ {∅}) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
2322simprbi 497 . . . . . 6 (∃𝑛(𝑛 ∈ (ω ∖ {∅}) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2421, 23sylbi 216 . . . . 5 (∃𝑛 ∈ (ω ∖ {∅})(𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2520, 24syl 17 . . . 4 (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
26 eqid 2736 . . . . . . 7 {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} = {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))}
278, 26bnj900 33541 . . . . . 6 (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} → ∅ ∈ dom 𝑓)
28 fveq2 6842 . . . . . . 7 (𝑖 = ∅ → (𝑓𝑖) = (𝑓‘∅))
2928ssiun2s 5008 . . . . . 6 (∅ ∈ dom 𝑓 → (𝑓‘∅) ⊆ 𝑖 ∈ dom 𝑓(𝑓𝑖))
3027, 29syl 17 . . . . 5 (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} → (𝑓‘∅) ⊆ 𝑖 ∈ dom 𝑓(𝑓𝑖))
31 ssiun2 5007 . . . . . 6 (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} → 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} 𝑖 ∈ dom 𝑓(𝑓𝑖))
326, 7, 8, 26bnj882 33538 . . . . . 6 trCl(𝑋, 𝐴, 𝑅) = 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} 𝑖 ∈ dom 𝑓(𝑓𝑖)
3331, 32sseqtrrdi 3995 . . . . 5 (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} → 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
3430, 33sstrd 3954 . . . 4 (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} → (𝑓‘∅) ⊆ trCl(𝑋, 𝐴, 𝑅))
3525, 34eqsstrrd 3983 . . 3 (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
3635exlimiv 1933 . 2 (∃𝑓 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
3717, 36syl 17 1 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2566  {cab 2713  wne 2943  wral 3064  wrex 3073  cdif 3907  wss 3910  c0 4282  {csn 4586   ciun 4954  dom cdm 5633  suc csuc 6319   Fn wfn 6491  cfv 6496  ωcom 7802  1oc1o 8405   predc-bnj14 33300   FrSe w-bnj15 33304   trClc-bnj18 33306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-bnj17 33299  df-bnj14 33301  df-bnj13 33303  df-bnj15 33305  df-bnj18 33307
This theorem is referenced by:  bnj1137  33607  bnj1136  33609  bnj1175  33616  bnj1177  33618  bnj1413  33647  bnj1408  33648  bnj1417  33653  bnj1442  33661  bnj1452  33664
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