Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj873 Structured version   Visualization version   GIF version

Theorem bnj873 31322
Description: Technical lemma for bnj69 31406. 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
bnj873.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj873.7 (𝜑′[𝑔 / 𝑓]𝜑)
bnj873.8 (𝜓′[𝑔 / 𝑓]𝜓)
Assertion
Ref Expression
bnj873 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
Distinct variable groups:   𝐷,𝑓,𝑔   𝑓,𝑛,𝑔   𝜑,𝑔   𝜓,𝑔
Allowed substitution hints:   𝜑(𝑓,𝑛)   𝜓(𝑓,𝑛)   𝐵(𝑓,𝑔,𝑛)   𝐷(𝑛)   𝜑′(𝑓,𝑔,𝑛)   𝜓′(𝑓,𝑔,𝑛)

Proof of Theorem bnj873
StepHypRef Expression
1 bnj873.4 . 2 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
2 nfv 1992 . . 3 𝑔𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
3 nfcv 2902 . . . 4 𝑓𝐷
4 nfv 1992 . . . . 5 𝑓 𝑔 Fn 𝑛
5 bnj873.7 . . . . . 6 (𝜑′[𝑔 / 𝑓]𝜑)
6 nfsbc1v 3596 . . . . . 6 𝑓[𝑔 / 𝑓]𝜑
75, 6nfxfr 1928 . . . . 5 𝑓𝜑′
8 bnj873.8 . . . . . 6 (𝜓′[𝑔 / 𝑓]𝜓)
9 nfsbc1v 3596 . . . . . 6 𝑓[𝑔 / 𝑓]𝜓
108, 9nfxfr 1928 . . . . 5 𝑓𝜓′
114, 7, 10nf3an 1980 . . . 4 𝑓(𝑔 Fn 𝑛𝜑′𝜓′)
123, 11nfrex 3145 . . 3 𝑓𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)
13 fneq1 6140 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑛𝑔 Fn 𝑛))
14 sbceq1a 3587 . . . . . 6 (𝑓 = 𝑔 → (𝜑[𝑔 / 𝑓]𝜑))
1514, 5syl6bbr 278 . . . . 5 (𝑓 = 𝑔 → (𝜑𝜑′))
16 sbceq1a 3587 . . . . . 6 (𝑓 = 𝑔 → (𝜓[𝑔 / 𝑓]𝜓))
1716, 8syl6bbr 278 . . . . 5 (𝑓 = 𝑔 → (𝜓𝜓′))
1813, 15, 173anbi123d 1548 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑔 Fn 𝑛𝜑′𝜓′)))
1918rexbidv 3190 . . 3 (𝑓 = 𝑔 → (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)))
202, 12, 19cbvab 2884 . 2 {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)} = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
211, 20eqtri 2782 1 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1072   = wceq 1632  {cab 2746  wrex 3051  [wsbc 3576   Fn wfn 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-fun 6051  df-fn 6052
This theorem is referenced by:  bnj849  31323  bnj893  31326
  Copyright terms: Public domain W3C validator