MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralrnmpt Structured version   Visualization version   GIF version

Theorem ralrnmpt 6334
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
ralrnmpt.1 𝐹 = (𝑥𝐴𝐵)
ralrnmpt.2 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
ralrnmpt (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝜒,𝑦   𝑦,𝐹   𝜓,𝑥
Allowed substitution hints:   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑦)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem ralrnmpt
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ralrnmpt.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
21fnmpt 5987 . . . 4 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
3 dfsbcq 3424 . . . . 5 (𝑤 = (𝐹𝑧) → ([𝑤 / 𝑦]𝜓[(𝐹𝑧) / 𝑦]𝜓))
43ralrn 6328 . . . 4 (𝐹 Fn 𝐴 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧𝐴 [(𝐹𝑧) / 𝑦]𝜓))
52, 4syl 17 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧𝐴 [(𝐹𝑧) / 𝑦]𝜓))
6 nfv 1840 . . . . 5 𝑤𝜓
7 nfsbc1v 3442 . . . . 5 𝑦[𝑤 / 𝑦]𝜓
8 sbceq1a 3433 . . . . 5 (𝑦 = 𝑤 → (𝜓[𝑤 / 𝑦]𝜓))
96, 7, 8cbvral 3159 . . . 4 (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓)
109bicomi 214 . . 3 (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
11 nfmpt1 4717 . . . . . . 7 𝑥(𝑥𝐴𝐵)
121, 11nfcxfr 2759 . . . . . 6 𝑥𝐹
13 nfcv 2761 . . . . . 6 𝑥𝑧
1412, 13nffv 6165 . . . . 5 𝑥(𝐹𝑧)
15 nfv 1840 . . . . 5 𝑥𝜓
1614, 15nfsbc 3444 . . . 4 𝑥[(𝐹𝑧) / 𝑦]𝜓
17 nfv 1840 . . . 4 𝑧[(𝐹𝑥) / 𝑦]𝜓
18 fveq2 6158 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1918sbceq1d 3427 . . . 4 (𝑧 = 𝑥 → ([(𝐹𝑧) / 𝑦]𝜓[(𝐹𝑥) / 𝑦]𝜓))
2016, 17, 19cbvral 3159 . . 3 (∀𝑧𝐴 [(𝐹𝑧) / 𝑦]𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓)
215, 10, 203bitr3g 302 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓))
221fvmpt2 6258 . . . . . 6 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
2322sbceq1d 3427 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓[𝐵 / 𝑦]𝜓))
24 ralrnmpt.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
2524sbcieg 3455 . . . . . 6 (𝐵𝑉 → ([𝐵 / 𝑦]𝜓𝜒))
2625adantl 482 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([𝐵 / 𝑦]𝜓𝜒))
2723, 26bitrd 268 . . . 4 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓𝜒))
2827ralimiaa 2947 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 ([(𝐹𝑥) / 𝑦]𝜓𝜒))
29 ralbi 3063 . . 3 (∀𝑥𝐴 ([(𝐹𝑥) / 𝑦]𝜓𝜒) → (∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓 ↔ ∀𝑥𝐴 𝜒))
3028, 29syl 17 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓 ↔ ∀𝑥𝐴 𝜒))
3121, 30bitrd 268 1 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  [wsbc 3422  cmpt 4683  ran crn 5085   Fn wfn 5852  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865
This theorem is referenced by:  rexrnmpt  6335  ac6num  9261  gsumwspan  17323  dfod2  17921  ordtbaslem  20932  ordtrest2lem  20947  cncmp  21135  comppfsc  21275  ptpjopn  21355  ordthmeolem  21544  tsmsfbas  21871  tsmsf1o  21888  prdsxmetlem  22113  prdsbl  22236  metdsf  22591  metdsge  22592  minveclem1  23135  minveclem3b  23139  minveclem6  23145  mbflimsup  23373  xrlimcnp  24629  minvecolem1  27618  minvecolem5  27625  minvecolem6  27626  ordtrest2NEWlem  29792  cvmsss2  31017  fin2so  33067  prdsbnd  33263  rrnequiv  33305
  Copyright terms: Public domain W3C validator