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Theorem divsfval 17495
Description: Value of the function in qusval 17490. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
Hypotheses
Ref Expression
ercpbl.r (𝜑 Er 𝑉)
ercpbl.v (𝜑𝑉𝑊)
ercpbl.f 𝐹 = (𝑥𝑉 ↦ [𝑥] )
Assertion
Ref Expression
divsfval (𝜑 → (𝐹𝐴) = [𝐴] )
Distinct variable groups:   𝑥,   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥
Allowed substitution hints:   𝐹(𝑥)   𝑊(𝑥)

Proof of Theorem divsfval
StepHypRef Expression
1 ercpbl.v . . . . 5 (𝜑𝑉𝑊)
2 ercpbl.r . . . . . 6 (𝜑 Er 𝑉)
32ecss 8751 . . . . 5 (𝜑 → [𝐴] 𝑉)
41, 3ssexd 5324 . . . 4 (𝜑 → [𝐴] ∈ V)
5 eceq1 8743 . . . . 5 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
6 ercpbl.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
75, 6fvmptg 6996 . . . 4 ((𝐴𝑉 ∧ [𝐴] ∈ V) → (𝐹𝐴) = [𝐴] )
84, 7sylan2 593 . . 3 ((𝐴𝑉𝜑) → (𝐹𝐴) = [𝐴] )
98expcom 414 . 2 (𝜑 → (𝐴𝑉 → (𝐹𝐴) = [𝐴] ))
106dmeqi 5904 . . . . . . . 8 dom 𝐹 = dom (𝑥𝑉 ↦ [𝑥] )
112ecss 8751 . . . . . . . . . . 11 (𝜑 → [𝑥] 𝑉)
121, 11ssexd 5324 . . . . . . . . . 10 (𝜑 → [𝑥] ∈ V)
1312ralrimivw 3150 . . . . . . . . 9 (𝜑 → ∀𝑥𝑉 [𝑥] ∈ V)
14 dmmptg 6241 . . . . . . . . 9 (∀𝑥𝑉 [𝑥] ∈ V → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1513, 14syl 17 . . . . . . . 8 (𝜑 → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1610, 15eqtrid 2784 . . . . . . 7 (𝜑 → dom 𝐹 = 𝑉)
1716eleq2d 2819 . . . . . 6 (𝜑 → (𝐴 ∈ dom 𝐹𝐴𝑉))
1817notbid 317 . . . . 5 (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴𝑉))
19 ndmfv 6926 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
2018, 19syl6bir 253 . . . 4 (𝜑 → (¬ 𝐴𝑉 → (𝐹𝐴) = ∅))
21 ecdmn0 8752 . . . . . 6 (𝐴 ∈ dom ↔ [𝐴] ≠ ∅)
22 erdm 8715 . . . . . . . . 9 ( Er 𝑉 → dom = 𝑉)
232, 22syl 17 . . . . . . . 8 (𝜑 → dom = 𝑉)
2423eleq2d 2819 . . . . . . 7 (𝜑 → (𝐴 ∈ dom 𝐴𝑉))
2524biimpd 228 . . . . . 6 (𝜑 → (𝐴 ∈ dom 𝐴𝑉))
2621, 25biimtrrid 242 . . . . 5 (𝜑 → ([𝐴] ≠ ∅ → 𝐴𝑉))
2726necon1bd 2958 . . . 4 (𝜑 → (¬ 𝐴𝑉 → [𝐴] = ∅))
2820, 27jcad 513 . . 3 (𝜑 → (¬ 𝐴𝑉 → ((𝐹𝐴) = ∅ ∧ [𝐴] = ∅)))
29 eqtr3 2758 . . 3 (((𝐹𝐴) = ∅ ∧ [𝐴] = ∅) → (𝐹𝐴) = [𝐴] )
3028, 29syl6 35 . 2 (𝜑 → (¬ 𝐴𝑉 → (𝐹𝐴) = [𝐴] ))
319, 30pm2.61d 179 1 (𝜑 → (𝐹𝐴) = [𝐴] )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  Vcvv 3474  c0 4322  cmpt 5231  dom cdm 5676  cfv 6543   Er wer 8702  [cec 8703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-er 8705  df-ec 8707
This theorem is referenced by:  ercpbllem  17496  qusaddvallem  17499  qusgrp2  18943  frgpmhm  19635  frgpup3lem  19647  qusring2  20151  qusrhm  20880
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