Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1234 Structured version   Visualization version   GIF version

Theorem bnj1234 30169
Description: Technical lemma for bnj60 30218. 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
bnj1234.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1234.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1234.4 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1234.5 𝐷 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
Assertion
Ref Expression
bnj1234 𝐶 = 𝐷
Distinct variable groups:   𝐵,𝑓,𝑔   𝑓,𝐺,𝑔   𝑔,𝑌   𝑓,𝑍   𝑓,𝑑,𝑔   𝑥,𝑓,𝑔
Allowed substitution hints:   𝐴(𝑥,𝑓,𝑔,𝑑)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑔,𝑑)   𝐷(𝑥,𝑓,𝑔,𝑑)   𝑅(𝑥,𝑓,𝑔,𝑑)   𝐺(𝑥,𝑑)   𝑌(𝑥,𝑓,𝑑)   𝑍(𝑥,𝑔,𝑑)

Proof of Theorem bnj1234
StepHypRef Expression
1 fneq1 5879 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑑𝑔 Fn 𝑑))
2 fveq1 6087 . . . . . . 7 (𝑓 = 𝑔 → (𝑓𝑥) = (𝑔𝑥))
3 reseq1 5298 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)))
43opeq2d 4341 . . . . . . . . 9 (𝑓 = 𝑔 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5 bnj1234.2 . . . . . . . . 9 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
6 bnj1234.4 . . . . . . . . 9 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
74, 5, 63eqtr4g 2668 . . . . . . . 8 (𝑓 = 𝑔𝑌 = 𝑍)
87fveq2d 6092 . . . . . . 7 (𝑓 = 𝑔 → (𝐺𝑌) = (𝐺𝑍))
92, 8eqeq12d 2624 . . . . . 6 (𝑓 = 𝑔 → ((𝑓𝑥) = (𝐺𝑌) ↔ (𝑔𝑥) = (𝐺𝑍)))
109ralbidv 2968 . . . . 5 (𝑓 = 𝑔 → (∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌) ↔ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
111, 10anbi12d 742 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))))
1211rexbidv 3033 . . 3 (𝑓 = 𝑔 → (∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))))
1312cbvabv 2733 . 2 {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))} = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
14 bnj1234.3 . 2 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
15 bnj1234.5 . 2 𝐷 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
1613, 14, 153eqtr4i 2641 1 𝐶 = 𝐷
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  {cab 2595  wral 2895  wrex 2896  cop 4130  cres 5030   Fn wfn 5785  cfv 5790   predc-bnj14 29841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-res 5040  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  bnj1245  30170  bnj1256  30171  bnj1259  30172  bnj1296  30177  bnj1311  30180
  Copyright terms: Public domain W3C validator