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Theorem resflem 5660
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 5660 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 3146 . . . . 5 (𝜑 → (𝑥𝐴𝑥𝑉))
3 resflem.1 . . . . . . 7 (𝜑𝐹:𝑉𝑋)
4 fdm 5353 . . . . . . 7 (𝐹:𝑉𝑋 → dom 𝐹 = 𝑉)
53, 4syl 14 . . . . . 6 (𝜑 → dom 𝐹 = 𝑉)
65eleq2d 2240 . . . . 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 2542 . 2 (𝜑 → ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑌))
12 ffun 5350 . . . 4 (𝐹:𝑉𝑋 → Fun 𝐹)
133, 12syl 14 . . 3 (𝜑 → Fun 𝐹)
14 ffvresb 5659 . . 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 1348  wcel 2141  wral 2448  wss 3121  dom cdm 4611  cres 4613  Fun wfun 5192  wf 5194  cfv 5198
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206
This theorem is referenced by:  bj-charfun  13842
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