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Theorem ralrn 5305
Description: Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
Hypothesis
Ref Expression
rexrn.1 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
Assertion
Ref Expression
ralrn (𝐹 Fn 𝐴 → (∀𝑥 ∈ ran 𝐹𝜑 ↔ ∀𝑦𝐴 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem ralrn
StepHypRef Expression
1 funfvex 5192 . . 3 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ V)
21funfni 4999 . 2 ((𝐹 Fn 𝐴𝑦𝐴) → (𝐹𝑦) ∈ V)
3 fvelrnb 5221 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = 𝑥))
4 eqcom 2042 . . . 4 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
54rexbii 2331 . . 3 (∃𝑦𝐴 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦))
63, 5syl6bb 185 . 2 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦)))
7 rexrn.1 . . 3 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
87adantl 262 . 2 ((𝐹 Fn 𝐴𝑥 = (𝐹𝑦)) → (𝜑𝜓))
92, 6, 8ralxfr2d 4196 1 (𝐹 Fn 𝐴 → (∀𝑥 ∈ ran 𝐹𝜑 ↔ ∀𝑦𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wcel 1393  wral 2306  wrex 2307  Vcvv 2557  ran crn 4346   Fn wfn 4897  cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by:  ralrnmpt  5309  cbvfo  5425  isoselem  5459
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