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Theorem ralrnmptw 7045
Description: A restricted quantifier over an image set. Version of ralrnmpt 7047 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
ralrnmptw.1 𝐹 = (𝑥𝐴𝐵)
ralrnmptw.2 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
ralrnmptw (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
Distinct variable groups:   𝑥,𝑦   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 ralrnmptw
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ralrnmptw.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
21fnmpt 6642 . . . 4 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
3 dfsbcq 3742 . . . . 5 (𝑤 = (𝐹𝑧) → ([𝑤 / 𝑦]𝜓[(𝐹𝑧) / 𝑦]𝜓))
43ralrn 7039 . . . 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 nfsbc1v 3760 . . . 4 𝑦[𝑤 / 𝑦]𝜓
7 nfv 1918 . . . 4 𝑤𝜓
8 sbceq2a 3752 . . . 4 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜓𝜓))
96, 7, 8cbvralw 3288 . . 3 (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
10 nfmpt1 5214 . . . . . . 7 𝑥(𝑥𝐴𝐵)
111, 10nfcxfr 2902 . . . . . 6 𝑥𝐹
12 nfcv 2904 . . . . . 6 𝑥𝑧
1311, 12nffv 6853 . . . . 5 𝑥(𝐹𝑧)
14 nfv 1918 . . . . 5 𝑥𝜓
1513, 14nfsbcw 3762 . . . 4 𝑥[(𝐹𝑧) / 𝑦]𝜓
16 nfv 1918 . . . 4 𝑧[(𝐹𝑥) / 𝑦]𝜓
17 fveq2 6843 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1817sbceq1d 3745 . . . 4 (𝑧 = 𝑥 → ([(𝐹𝑧) / 𝑦]𝜓[(𝐹𝑥) / 𝑦]𝜓))
1915, 16, 18cbvralw 3288 . . 3 (∀𝑧𝐴 [(𝐹𝑧) / 𝑦]𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓)
205, 9, 193bitr3g 313 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓))
211fvmpt2 6960 . . . . . 6 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
2221sbceq1d 3745 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓[𝐵 / 𝑦]𝜓))
23 ralrnmptw.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
2423sbcieg 3780 . . . . . 6 (𝐵𝑉 → ([𝐵 / 𝑦]𝜓𝜒))
2524adantl 483 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([𝐵 / 𝑦]𝜓𝜒))
2622, 25bitrd 279 . . . 4 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓𝜒))
2726ralimiaa 3082 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 ([(𝐹𝑥) / 𝑦]𝜓𝜒))
28 ralbi 3103 . . 3 (∀𝑥𝐴 ([(𝐹𝑥) / 𝑦]𝜓𝜒) → (∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓 ↔ ∀𝑥𝐴 𝜒))
2927, 28syl 17 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓 ↔ ∀𝑥𝐴 𝜒))
3020, 29bitrd 279 1 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  [wsbc 3740  cmpt 5189  ran crn 5635   Fn wfn 6492  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  rexrnmptw  7046  ac6num  10420  gsumwspan  18661  dfod2  19351  ordtbaslem  22555  ordtrest2lem  22570  cncmp  22759  comppfsc  22899  ptpjopn  22979  ordthmeolem  23168  tsmsfbas  23495  tsmsf1o  23512  prdsxmetlem  23737  prdsbl  23863  metdsf  24227  metdsge  24228  minveclem1  24804  minveclem3b  24808  minveclem6  24814  mbflimsup  25046  xrlimcnp  26334  minvecolem1  29858  minvecolem5  29865  minvecolem6  29866  ordtrest2NEWlem  32560  cvmsss2  33925  fin2so  36111  prdsbnd  36298  rrnequiv  36340  ralrnmpt3  43574
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