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Theorem resflem 5456
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 5456 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 3024 . . . . 5 (𝜑 → (𝑥𝐴𝑥𝑉))
3 resflem.1 . . . . . . 7 (𝜑𝐹:𝑉𝑋)
4 fdm 5160 . . . . . . 7 (𝐹:𝑉𝑋 → dom 𝐹 = 𝑉)
53, 4syl 14 . . . . . 6 (𝜑 → dom 𝐹 = 𝑉)
65eleq2d 2157 . . . . 5 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝑉))
72, 6sylibrd 167 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ dom 𝐹))
8 resflem.3 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝑌)
98ex 113 . . . 4 (𝜑 → (𝑥𝐴 → (𝐹𝑥) ∈ 𝑌))
107, 9jcad 301 . . 3 (𝜑 → (𝑥𝐴 → (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑌)))
1110ralrimiv 2445 . 2 (𝜑 → ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑌))
12 ffun 5158 . . . 4 (𝐹:𝑉𝑋 → Fun 𝐹)
133, 12syl 14 . . 3 (𝜑 → Fun 𝐹)
14 ffvresb 5455 . . 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 165 1 (𝜑 → (𝐹𝐴):𝐴𝑌)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  wss 2999  dom cdm 4436  cres 4438  Fun wfun 5004  wf 5006  cfv 5010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-fv 5018
This theorem is referenced by: (None)
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