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Theorem ralrnmpt 6532
 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 6181 . . . 4 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
3 dfsbcq 3578 . . . . 5 (𝑤 = (𝐹𝑧) → ([𝑤 / 𝑦]𝜓[(𝐹𝑧) / 𝑦]𝜓))
43ralrn 6526 . . . 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 1992 . . . . 5 𝑤𝜓
7 nfsbc1v 3596 . . . . 5 𝑦[𝑤 / 𝑦]𝜓
8 sbceq1a 3587 . . . . 5 (𝑦 = 𝑤 → (𝜓[𝑤 / 𝑦]𝜓))
96, 7, 8cbvral 3306 . . . 4 (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓)
109bicomi 214 . . 3 (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
11 nfmpt1 4899 . . . . . . 7 𝑥(𝑥𝐴𝐵)
121, 11nfcxfr 2900 . . . . . 6 𝑥𝐹
13 nfcv 2902 . . . . . 6 𝑥𝑧
1412, 13nffv 6360 . . . . 5 𝑥(𝐹𝑧)
15 nfv 1992 . . . . 5 𝑥𝜓
1614, 15nfsbc 3598 . . . 4 𝑥[(𝐹𝑧) / 𝑦]𝜓
17 nfv 1992 . . . 4 𝑧[(𝐹𝑥) / 𝑦]𝜓
18 fveq2 6353 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1918sbceq1d 3581 . . . 4 (𝑧 = 𝑥 → ([(𝐹𝑧) / 𝑦]𝜓[(𝐹𝑥) / 𝑦]𝜓))
2016, 17, 19cbvral 3306 . . 3 (∀𝑧𝐴 [(𝐹𝑧) / 𝑦]𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓)
215, 10, 203bitr3g 302 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓))
221fvmpt2 6454 . . . . . 6 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
2322sbceq1d 3581 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓[𝐵 / 𝑦]𝜓))
24 ralrnmpt.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
2524sbcieg 3609 . . . . . 6 (𝐵𝑉 → ([𝐵 / 𝑦]𝜓𝜒))
2625adantl 473 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([𝐵 / 𝑦]𝜓𝜒))
2723, 26bitrd 268 . . . 4 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓𝜒))
2827ralimiaa 3089 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 ([(𝐹𝑥) / 𝑦]𝜓𝜒))
29 ralbi 3206 . . 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 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  [wsbc 3576   ↦ cmpt 4881  ran crn 5267   Fn wfn 6044  ‘cfv 6049 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-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 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-eu 2611  df-mo 2612  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-csb 3675  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-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057 This theorem is referenced by:  rexrnmpt  6533  ac6num  9513  gsumwspan  17604  dfod2  18201  ordtbaslem  21214  ordtrest2lem  21229  cncmp  21417  comppfsc  21557  ptpjopn  21637  ordthmeolem  21826  tsmsfbas  22152  tsmsf1o  22169  prdsxmetlem  22394  prdsbl  22517  metdsf  22872  metdsge  22873  minveclem1  23415  minveclem3b  23419  minveclem6  23425  mbflimsup  23652  xrlimcnp  24915  minvecolem1  28060  minvecolem5  28067  minvecolem6  28068  ordtrest2NEWlem  30298  cvmsss2  31584  fin2so  33727  prdsbnd  33923  rrnequiv  33965  ralrnmpt3  39991
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