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Theorem divsfval 16188
Description: Value of the function in qusval 16183. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ercpbl.r (𝜑 Er 𝑉)
ercpbl.v (𝜑𝑉 ∈ V)
ercpbl.f 𝐹 = (𝑥𝑉 ↦ [𝑥] )
Assertion
Ref Expression
divsfval (𝜑 → (𝐹𝐴) = [𝐴] )
Distinct variable groups:   𝑥,   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem divsfval
StepHypRef Expression
1 ercpbl.v . . . . 5 (𝜑𝑉 ∈ V)
2 ercpbl.r . . . . . 6 (𝜑 Er 𝑉)
32ecss 7773 . . . . 5 (𝜑 → [𝐴] 𝑉)
41, 3ssexd 4796 . . . 4 (𝜑 → [𝐴] ∈ V)
5 eceq1 7767 . . . . 5 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
6 ercpbl.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
75, 6fvmptg 6267 . . . 4 ((𝐴𝑉 ∧ [𝐴] ∈ V) → (𝐹𝐴) = [𝐴] )
84, 7sylan2 491 . . 3 ((𝐴𝑉𝜑) → (𝐹𝐴) = [𝐴] )
98expcom 451 . 2 (𝜑 → (𝐴𝑉 → (𝐹𝐴) = [𝐴] ))
106dmeqi 5314 . . . . . . . 8 dom 𝐹 = dom (𝑥𝑉 ↦ [𝑥] )
112ecss 7773 . . . . . . . . . . 11 (𝜑 → [𝑥] 𝑉)
121, 11ssexd 4796 . . . . . . . . . 10 (𝜑 → [𝑥] ∈ V)
1312ralrimivw 2964 . . . . . . . . 9 (𝜑 → ∀𝑥𝑉 [𝑥] ∈ V)
14 dmmptg 5620 . . . . . . . . 9 (∀𝑥𝑉 [𝑥] ∈ V → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1513, 14syl 17 . . . . . . . 8 (𝜑 → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1610, 15syl5eq 2666 . . . . . . 7 (𝜑 → dom 𝐹 = 𝑉)
1716eleq2d 2685 . . . . . 6 (𝜑 → (𝐴 ∈ dom 𝐹𝐴𝑉))
1817notbid 308 . . . . 5 (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴𝑉))
19 ndmfv 6205 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
2018, 19syl6bir 244 . . . 4 (𝜑 → (¬ 𝐴𝑉 → (𝐹𝐴) = ∅))
21 ecdmn0 7774 . . . . . 6 (𝐴 ∈ dom ↔ [𝐴] ≠ ∅)
22 erdm 7737 . . . . . . . . 9 ( Er 𝑉 → dom = 𝑉)
232, 22syl 17 . . . . . . . 8 (𝜑 → dom = 𝑉)
2423eleq2d 2685 . . . . . . 7 (𝜑 → (𝐴 ∈ dom 𝐴𝑉))
2524biimpd 219 . . . . . 6 (𝜑 → (𝐴 ∈ dom 𝐴𝑉))
2621, 25syl5bir 233 . . . . 5 (𝜑 → ([𝐴] ≠ ∅ → 𝐴𝑉))
2726necon1bd 2809 . . . 4 (𝜑 → (¬ 𝐴𝑉 → [𝐴] = ∅))
2820, 27jcad 555 . . 3 (𝜑 → (¬ 𝐴𝑉 → ((𝐹𝐴) = ∅ ∧ [𝐴] = ∅)))
29 eqtr3 2641 . . 3 (((𝐹𝐴) = ∅ ∧ [𝐴] = ∅) → (𝐹𝐴) = [𝐴] )
3028, 29syl6 35 . 2 (𝜑 → (¬ 𝐴𝑉 → (𝐹𝐴) = [𝐴] ))
319, 30pm2.61d 170 1 (𝜑 → (𝐹𝐴) = [𝐴] )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  Vcvv 3195  c0 3907  cmpt 4720  dom cdm 5104  cfv 5876   Er wer 7724  [cec 7725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-er 7727  df-ec 7729
This theorem is referenced by:  ercpbllem  16189  qusaddvallem  16192  qusgrp2  17514  frgpmhm  18159  frgpup3lem  18171  qusring2  18601  qusrhm  19218
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