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Theorem ralrnmptw 7095
Description: A restricted quantifier over an image set. Version of ralrnmpt 7097 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 6690 . . . 4 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
3 dfsbcq 3779 . . . . 5 (𝑤 = (𝐹𝑧) → ([𝑤 / 𝑦]𝜓[(𝐹𝑧) / 𝑦]𝜓))
43ralrn 7089 . . . 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 3797 . . . 4 𝑦[𝑤 / 𝑦]𝜓
7 nfv 1917 . . . 4 𝑤𝜓
8 sbceq2a 3789 . . . 4 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜓𝜓))
96, 7, 8cbvralw 3303 . . 3 (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
10 nfmpt1 5256 . . . . . . 7 𝑥(𝑥𝐴𝐵)
111, 10nfcxfr 2901 . . . . . 6 𝑥𝐹
12 nfcv 2903 . . . . . 6 𝑥𝑧
1311, 12nffv 6901 . . . . 5 𝑥(𝐹𝑧)
14 nfv 1917 . . . . 5 𝑥𝜓
1513, 14nfsbcw 3799 . . . 4 𝑥[(𝐹𝑧) / 𝑦]𝜓
16 nfv 1917 . . . 4 𝑧[(𝐹𝑥) / 𝑦]𝜓
17 fveq2 6891 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1817sbceq1d 3782 . . . 4 (𝑧 = 𝑥 → ([(𝐹𝑧) / 𝑦]𝜓[(𝐹𝑥) / 𝑦]𝜓))
1915, 16, 18cbvralw 3303 . . 3 (∀𝑧𝐴 [(𝐹𝑧) / 𝑦]𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓)
205, 9, 193bitr3g 312 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 [(𝐹𝑥) / 𝑦]𝜓))
211fvmpt2 7009 . . . . . 6 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
2221sbceq1d 3782 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([(𝐹𝑥) / 𝑦]𝜓[𝐵 / 𝑦]𝜓))
23 ralrnmptw.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
2423sbcieg 3817 . . . . . 6 (𝐵𝑉 → ([𝐵 / 𝑦]𝜓𝜒))
2524adantl 482 . . . . 5 ((𝑥𝐴𝐵𝑉) → ([𝐵 / 𝑦]𝜓𝜒))
2622, 25bitrd 278 . . . 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 278 1 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  [wsbc 3777  cmpt 5231  ran crn 5677   Fn wfn 6538  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  rexrnmptw  7096  ac6num  10476  gsumwspan  18729  dfod2  19434  ordtbaslem  22699  ordtrest2lem  22714  cncmp  22903  comppfsc  23043  ptpjopn  23123  ordthmeolem  23312  tsmsfbas  23639  tsmsf1o  23656  prdsxmetlem  23881  prdsbl  24007  metdsf  24371  metdsge  24372  minveclem1  24948  minveclem3b  24952  minveclem6  24958  mbflimsup  25190  xrlimcnp  26480  minvecolem1  30165  minvecolem5  30172  minvecolem6  30173  ordtrest2NEWlem  32971  cvmsss2  34334  fin2so  36561  prdsbnd  36747  rrnequiv  36789  ralrnmpt3  44042
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