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Theorem resflem 5577
 Description: A lemma to bound the range of a restriction. The conclusion would also hold with (𝑋 ∩ 𝑌) 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 5577 where (𝑋 ∩ 𝑌) 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 3091 . . . . 5 (𝜑 → (𝑥𝐴𝑥𝑉))
3 resflem.1 . . . . . . 7 (𝜑𝐹:𝑉𝑋)
4 fdm 5273 . . . . . . 7 (𝐹:𝑉𝑋 → dom 𝐹 = 𝑉)
53, 4syl 14 . . . . . 6 (𝜑 → dom 𝐹 = 𝑉)
65eleq2d 2207 . . . . 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 2502 . 2 (𝜑 → ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑌))
12 ffun 5270 . . . 4 (𝐹:𝑉𝑋 → Fun 𝐹)
133, 12syl 14 . . 3 (𝜑 → Fun 𝐹)
14 ffvresb 5576 . . 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 1331   ∈ wcel 1480  ∀wral 2414   ⊆ wss 3066  dom cdm 4534   ↾ cres 4536  Fun wfun 5112  ⟶wf 5114  ‘cfv 5118 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126 This theorem is referenced by: (None)
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