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Theorem ralrnmpt 6258
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 5916 . . . 4 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
3 dfsbcq 3400 . . . . 5 (𝑤 = (𝐹𝑧) → ([𝑤 / 𝑦]𝜓[(𝐹𝑧) / 𝑦]𝜓))
43ralrn 6252 . . . 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 1829 . . . . 5 𝑤𝜓
7 nfsbc1v 3418 . . . . 5 𝑦[𝑤 / 𝑦]𝜓
8 sbceq1a 3409 . . . . 5 (𝑦 = 𝑤 → (𝜓[𝑤 / 𝑦]𝜓))
96, 7, 8cbvral 3139 . . . 4 (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓)
109bicomi 212 . . 3 (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
11 nfmpt1 4666 . . . . . . 7 𝑥(𝑥𝐴𝐵)
121, 11nfcxfr 2745 . . . . . 6 𝑥𝐹
13 nfcv 2747 . . . . . 6 𝑥𝑧
1412, 13nffv 6092 . . . . 5 𝑥(𝐹𝑧)
15 nfv 1829 . . . . 5 𝑥𝜓
1614, 15nfsbc 3420 . . . 4 𝑥[(𝐹𝑧) / 𝑦]𝜓
17 nfv 1829 . . . 4 𝑧[(𝐹𝑥) / 𝑦]𝜓
18 fveq2 6085 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1918sbceq1d 3403 . . . 4 (𝑧 = 𝑥 → ([(𝐹𝑧) / 𝑦]𝜓[(𝐹𝑥) / 𝑦]𝜓))
2016, 17, 19cbvral 3139 . . 3 (∀𝑧𝐴 [(𝐹𝑧) / 𝑦]𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓)
215, 10, 203bitr3g 300 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓))
221fvmpt2 6182 . . . . . 6 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
2322sbceq1d 3403 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓[𝐵 / 𝑦]𝜓))
24 ralrnmpt.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
2524sbcieg 3431 . . . . . 6 (𝐵𝑉 → ([𝐵 / 𝑦]𝜓𝜒))
2625adantl 480 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([𝐵 / 𝑦]𝜓𝜒))
2723, 26bitrd 266 . . . 4 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓𝜒))
2827ralimiaa 2931 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 ([(𝐹𝑥) / 𝑦]𝜓𝜒))
29 ralbi 3046 . . 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 266 1 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2892  [wsbc 3398  cmpt 4634  ran crn 5026   Fn wfn 5782  cfv 5787
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-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825
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-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-fv 5795
This theorem is referenced by:  rexrnmpt  6259  ac6num  9158  gsumwspan  17149  dfod2  17747  ordtbaslem  20741  ordtrest2lem  20756  cncmp  20944  comppfsc  21084  ptpjopn  21164  ordthmeolem  21353  tsmsfbas  21680  tsmsf1o  21697  prdsxmetlem  21921  prdsbl  22044  metdsf  22387  metdsge  22388  minveclem1  22917  minveclem3b  22921  minveclem6  22927  mbflimsup  23153  xrlimcnp  24409  minvecolem1  26917  minvecolem5  26924  minvecolem6  26925  ordtrest2NEWlem  29099  cvmsss2  30313  fin2so  32366  prdsbnd  32562  rrnequiv  32604
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