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Theorem ralrnmpt 6716
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 6348 . . . 4 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
3 dfsbcq 3703 . . . . 5 (𝑤 = (𝐹𝑧) → ([𝑤 / 𝑦]𝜓[(𝐹𝑧) / 𝑦]𝜓))
43ralrn 6710 . . . 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 1890 . . . . 5 𝑤𝜓
7 nfsbc1v 3721 . . . . 5 𝑦[𝑤 / 𝑦]𝜓
8 sbceq1a 3712 . . . . 5 (𝑦 = 𝑤 → (𝜓[𝑤 / 𝑦]𝜓))
96, 7, 8cbvral 3396 . . . 4 (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓)
109bicomi 225 . . 3 (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
11 nfmpt1 5052 . . . . . . 7 𝑥(𝑥𝐴𝐵)
121, 11nfcxfr 2945 . . . . . 6 𝑥𝐹
13 nfcv 2947 . . . . . 6 𝑥𝑧
1412, 13nffv 6540 . . . . 5 𝑥(𝐹𝑧)
15 nfv 1890 . . . . 5 𝑥𝜓
1614, 15nfsbc 3723 . . . 4 𝑥[(𝐹𝑧) / 𝑦]𝜓
17 nfv 1890 . . . 4 𝑧[(𝐹𝑥) / 𝑦]𝜓
18 fveq2 6530 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1918sbceq1d 3706 . . . 4 (𝑧 = 𝑥 → ([(𝐹𝑧) / 𝑦]𝜓[(𝐹𝑥) / 𝑦]𝜓))
2016, 17, 19cbvral 3396 . . 3 (∀𝑧𝐴 [(𝐹𝑧) / 𝑦]𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓)
215, 10, 203bitr3g 314 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓))
221fvmpt2 6636 . . . . . 6 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
2322sbceq1d 3706 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓[𝐵 / 𝑦]𝜓))
24 ralrnmpt.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
2524sbcieg 3734 . . . . . 6 (𝐵𝑉 → ([𝐵 / 𝑦]𝜓𝜒))
2625adantl 482 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([𝐵 / 𝑦]𝜓𝜒))
2723, 26bitrd 280 . . . 4 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓𝜒))
2827ralimiaa 3124 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 ([(𝐹𝑥) / 𝑦]𝜓𝜒))
29 ralbi 3132 . . 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 280 1 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1520  wcel 2079  wral 3103  [wsbc 3701  cmpt 5035  ran crn 5436   Fn wfn 6212  cfv 6217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-fv 6225
This theorem is referenced by:  rexrnmpt  6717  ac6num  9736  gsumwspan  17810  dfod2  18409  ordtbaslem  21468  ordtrest2lem  21483  cncmp  21672  comppfsc  21812  ptpjopn  21892  ordthmeolem  22081  tsmsfbas  22407  tsmsf1o  22424  prdsxmetlem  22649  prdsbl  22772  metdsf  23127  metdsge  23128  minveclem1  23698  minveclem3b  23702  minveclem6  23708  mbflimsup  23938  xrlimcnp  25216  minvecolem1  28330  minvecolem5  28337  minvecolem6  28338  ordtrest2NEWlem  30738  cvmsss2  32085  fin2so  34356  prdsbnd  34549  rrnequiv  34591  ralrnmpt3  41025
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