ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resflem GIF version

Theorem resflem 5632
Description: A lemma to bound the range of a restriction. The conclusion would also hold with (𝑋𝑌) in place of 𝑌 (provided 𝑥 does not occur in 𝑋). If that stronger result is needed, it is however simpler to use the instance of resflem 5632 where (𝑋𝑌) is substituted for 𝑌 (in both the conclusion and the third hypothesis). (Contributed by BJ, 4-Jul-2022.)
Hypotheses
Ref Expression
resflem.1 (𝜑𝐹:𝑉𝑋)
resflem.2 (𝜑𝐴𝑉)
resflem.3 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝑌)
Assertion
Ref Expression
resflem (𝜑 → (𝐹𝐴):𝐴𝑌)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐹   𝑥,𝑌
Allowed substitution hints:   𝑉(𝑥)   𝑋(𝑥)

Proof of Theorem resflem
StepHypRef Expression
1 resflem.2 . . . . . 6 (𝜑𝐴𝑉)
21sseld 3127 . . . . 5 (𝜑 → (𝑥𝐴𝑥𝑉))
3 resflem.1 . . . . . . 7 (𝜑𝐹:𝑉𝑋)
4 fdm 5326 . . . . . . 7 (𝐹:𝑉𝑋 → dom 𝐹 = 𝑉)
53, 4syl 14 . . . . . 6 (𝜑 → dom 𝐹 = 𝑉)
65eleq2d 2227 . . . . 5 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝑉))
72, 6sylibrd 168 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ dom 𝐹))
8 resflem.3 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝑌)
98ex 114 . . . 4 (𝜑 → (𝑥𝐴 → (𝐹𝑥) ∈ 𝑌))
107, 9jcad 305 . . 3 (𝜑 → (𝑥𝐴 → (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑌)))
1110ralrimiv 2529 . 2 (𝜑 → ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑌))
12 ffun 5323 . . . 4 (𝐹:𝑉𝑋 → Fun 𝐹)
133, 12syl 14 . . 3 (𝜑 → Fun 𝐹)
14 ffvresb 5631 . . 3 (Fun 𝐹 → ((𝐹𝐴):𝐴𝑌 ↔ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑌)))
1513, 14syl 14 . 2 (𝜑 → ((𝐹𝐴):𝐴𝑌 ↔ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑌)))
1611, 15mpbird 166 1 (𝜑 → (𝐹𝐴):𝐴𝑌)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wcel 2128  wral 2435  wss 3102  dom cdm 4587  cres 4589  Fun wfun 5165  wf 5167  cfv 5171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-fv 5179
This theorem is referenced by:  bj-charfun  13424
  Copyright terms: Public domain W3C validator