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Theorem respreima 5645
Description: The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
respreima (Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))

Proof of Theorem respreima
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funfn 5247 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elin 3319 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥𝐵𝑥 ∈ dom 𝐹))
3 ancom 266 . . . . . . . . 9 ((𝑥𝐵𝑥 ∈ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹𝑥𝐵))
42, 3bitri 184 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹𝑥𝐵))
54anbi1i 458 . . . . . . 7 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴))
6 fvres 5540 . . . . . . . . . 10 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
76eleq1d 2246 . . . . . . . . 9 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
87adantl 277 . . . . . . . 8 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
98pm5.32i 454 . . . . . . 7 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
105, 9bitri 184 . . . . . 6 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
1110a1i 9 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴)))
12 an32 562 . . . . 5 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵))
1311, 12bitrdi 196 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
14 fnfun 5314 . . . . . . . 8 (𝐹 Fn dom 𝐹 → Fun 𝐹)
15 funres 5258 . . . . . . . 8 (Fun 𝐹 → Fun (𝐹𝐵))
1614, 15syl 14 . . . . . . 7 (𝐹 Fn dom 𝐹 → Fun (𝐹𝐵))
17 dmres 4929 . . . . . . 7 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1816, 17jctir 313 . . . . . 6 (𝐹 Fn dom 𝐹 → (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)))
19 df-fn 5220 . . . . . 6 ((𝐹𝐵) Fn (𝐵 ∩ dom 𝐹) ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)))
2018, 19sylibr 134 . . . . 5 (𝐹 Fn dom 𝐹 → (𝐹𝐵) Fn (𝐵 ∩ dom 𝐹))
21 elpreima 5636 . . . . 5 ((𝐹𝐵) Fn (𝐵 ∩ dom 𝐹) → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴)))
2220, 21syl 14 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴)))
23 elin 3319 . . . . 5 (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ (𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵))
24 elpreima 5636 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴)))
2524anbi1d 465 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2623, 25bitrid 192 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2713, 22, 263bitr4d 220 . . 3 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
281, 27sylbi 121 . 2 (Fun 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
2928eqrdv 2175 1 (Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  cin 3129  ccnv 4626  dom cdm 4627  cres 4629  cima 4630  Fun wfun 5211   Fn wfn 5212  cfv 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225
This theorem is referenced by: (None)
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