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Theorem bnj18eq1 31377
Description: Equality theorem for transitive closure. (Contributed by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Assertion
Ref Expression
bnj18eq1 (𝑋 = 𝑌 → trCl(𝑋, 𝐴, 𝑅) = trCl(𝑌, 𝐴, 𝑅))

Proof of Theorem bnj18eq1
Dummy variables 𝑓 𝑖 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj602 31365 . . . . . . . . . . 11 (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
21eqeq2d 2775 . . . . . . . . . 10 (𝑋 = 𝑌 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑌, 𝐴, 𝑅)))
323anbi2d 1565 . . . . . . . . 9 (𝑋 = 𝑌 → ((𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
43rexbidv 3199 . . . . . . . 8 (𝑋 = 𝑌 → (∃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(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
54abbidv 2884 . . . . . . 7 (𝑋 = 𝑌 → {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(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))})
65eleq2d 2830 . . . . . 6 (𝑋 = 𝑌 → (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 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))}))
76anbi1d 623 . . . . 5 (𝑋 = 𝑌 → ((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)))} ∧ 𝑥 𝑖 ∈ dom 𝑓(𝑓𝑖)) ↔ (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} ∧ 𝑥 𝑖 ∈ dom 𝑓(𝑓𝑖))))
87rexbidv2 3195 . . . 4 (𝑋 = 𝑌 → (∃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)))}𝑥 𝑖 ∈ dom 𝑓(𝑓𝑖) ↔ ∃𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))}𝑥 𝑖 ∈ dom 𝑓(𝑓𝑖)))
98abbidv 2884 . . 3 (𝑋 = 𝑌 → {𝑥 ∣ ∃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)))}𝑥 𝑖 ∈ dom 𝑓(𝑓𝑖)} = {𝑥 ∣ ∃𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))}𝑥 𝑖 ∈ dom 𝑓(𝑓𝑖)})
10 df-iun 4678 . . 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)))} 𝑖 ∈ dom 𝑓(𝑓𝑖) = {𝑥 ∣ ∃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)))}𝑥 𝑖 ∈ dom 𝑓(𝑓𝑖)}
11 df-iun 4678 . . 3 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} 𝑖 ∈ dom 𝑓(𝑓𝑖) = {𝑥 ∣ ∃𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))}𝑥 𝑖 ∈ dom 𝑓(𝑓𝑖)}
129, 10, 113eqtr4g 2824 . 2 (𝑋 = 𝑌 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(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} 𝑖 ∈ dom 𝑓(𝑓𝑖))
13 df-bnj18 31144 . 2 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)
14 df-bnj18 31144 . 2 trCl(𝑌, 𝐴, 𝑅) = 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑌, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} 𝑖 ∈ dom 𝑓(𝑓𝑖)
1512, 13, 143eqtr4g 2824 1 (𝑋 = 𝑌 → trCl(𝑋, 𝐴, 𝑅) = trCl(𝑌, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1107   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  cdif 3729  c0 4079  {csn 4334   ciun 4676  dom cdm 5277  suc csuc 5910   Fn wfn 6063  cfv 6068  ωcom 7263   predc-bnj14 31137   trClc-bnj18 31143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-iun 4678  df-br 4810  df-bnj14 31138  df-bnj18 31144
This theorem is referenced by:  bnj1137  31443
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