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Theorem ralrnmptw 7096
Description: A restricted quantifier over an image set. Version of ralrnmpt 7098 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2370. (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 6691 . . . 4 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
3 dfsbcq 3780 . . . . 5 (𝑤 = (𝐹𝑧) → ([𝑤 / 𝑦]𝜓[(𝐹𝑧) / 𝑦]𝜓))
43ralrn 7090 . . . 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 3798 . . . 4 𝑦[𝑤 / 𝑦]𝜓
7 nfv 1916 . . . 4 𝑤𝜓
8 sbceq2a 3790 . . . 4 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜓𝜓))
96, 7, 8cbvralw 3302 . . 3 (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
10 nfmpt1 5257 . . . . . . 7 𝑥(𝑥𝐴𝐵)
111, 10nfcxfr 2900 . . . . . 6 𝑥𝐹
12 nfcv 2902 . . . . . 6 𝑥𝑧
1311, 12nffv 6902 . . . . 5 𝑥(𝐹𝑧)
14 nfv 1916 . . . . 5 𝑥𝜓
1513, 14nfsbcw 3800 . . . 4 𝑥[(𝐹𝑧) / 𝑦]𝜓
16 nfv 1916 . . . 4 𝑧[(𝐹𝑥) / 𝑦]𝜓
17 fveq2 6892 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1817sbceq1d 3783 . . . 4 (𝑧 = 𝑥 → ([(𝐹𝑧) / 𝑦]𝜓[(𝐹𝑥) / 𝑦]𝜓))
1915, 16, 18cbvralw 3302 . . 3 (∀𝑧𝐴 [(𝐹𝑧) / 𝑦]𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓)
205, 9, 193bitr3g 312 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓))
211fvmpt2 7010 . . . . . 6 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
2221sbceq1d 3783 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓[𝐵 / 𝑦]𝜓))
23 ralrnmptw.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
2423sbcieg 3818 . . . . . 6 (𝐵𝑉 → ([𝐵 / 𝑦]𝜓𝜒))
2524adantl 481 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([𝐵 / 𝑦]𝜓𝜒))
2622, 25bitrd 278 . . . 4 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓𝜒))
2726ralimiaa 3081 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 ([(𝐹𝑥) / 𝑦]𝜓𝜒))
28 ralbi 3102 . . 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 278 1 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wral 3060  [wsbc 3778  cmpt 5232  ran crn 5678   Fn wfn 6539  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  rexrnmptw  7097  ac6num  10477  gsumwspan  18764  dfod2  19474  ordtbaslem  22913  ordtrest2lem  22928  cncmp  23117  comppfsc  23257  ptpjopn  23337  ordthmeolem  23526  tsmsfbas  23853  tsmsf1o  23870  prdsxmetlem  24095  prdsbl  24221  metdsf  24585  metdsge  24586  minveclem1  25173  minveclem3b  25177  minveclem6  25183  mbflimsup  25416  xrlimcnp  26706  minvecolem1  30391  minvecolem5  30398  minvecolem6  30399  ordtrest2NEWlem  33197  cvmsss2  34560  fin2so  36779  prdsbnd  36965  rrnequiv  37007  ralrnmpt3  44263
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