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Theorem divsfval 17258
Description: Value of the function in qusval 17253. (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 8544 . . . . 5 (𝜑 → [𝐴] 𝑉)
41, 3ssexd 5248 . . . 4 (𝜑 → [𝐴] ∈ V)
5 eceq1 8536 . . . . 5 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
6 ercpbl.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
75, 6fvmptg 6873 . . . 4 ((𝐴𝑉 ∧ [𝐴] ∈ V) → (𝐹𝐴) = [𝐴] )
84, 7sylan2 593 . . 3 ((𝐴𝑉𝜑) → (𝐹𝐴) = [𝐴] )
98expcom 414 . 2 (𝜑 → (𝐴𝑉 → (𝐹𝐴) = [𝐴] ))
106dmeqi 5813 . . . . . . . 8 dom 𝐹 = dom (𝑥𝑉 ↦ [𝑥] )
112ecss 8544 . . . . . . . . . . 11 (𝜑 → [𝑥] 𝑉)
121, 11ssexd 5248 . . . . . . . . . 10 (𝜑 → [𝑥] ∈ V)
1312ralrimivw 3104 . . . . . . . . 9 (𝜑 → ∀𝑥𝑉 [𝑥] ∈ V)
14 dmmptg 6145 . . . . . . . . 9 (∀𝑥𝑉 [𝑥] ∈ V → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1513, 14syl 17 . . . . . . . 8 (𝜑 → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1610, 15eqtrid 2790 . . . . . . 7 (𝜑 → dom 𝐹 = 𝑉)
1716eleq2d 2824 . . . . . 6 (𝜑 → (𝐴 ∈ dom 𝐹𝐴𝑉))
1817notbid 318 . . . . 5 (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴𝑉))
19 ndmfv 6804 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
2018, 19syl6bir 253 . . . 4 (𝜑 → (¬ 𝐴𝑉 → (𝐹𝐴) = ∅))
21 ecdmn0 8545 . . . . . 6 (𝐴 ∈ dom ↔ [𝐴] ≠ ∅)
22 erdm 8508 . . . . . . . . 9 ( Er 𝑉 → dom = 𝑉)
232, 22syl 17 . . . . . . . 8 (𝜑 → dom = 𝑉)
2423eleq2d 2824 . . . . . . 7 (𝜑 → (𝐴 ∈ dom 𝐴𝑉))
2524biimpd 228 . . . . . 6 (𝜑 → (𝐴 ∈ dom 𝐴𝑉))
2621, 25syl5bir 242 . . . . 5 (𝜑 → ([𝐴] ≠ ∅ → 𝐴𝑉))
2726necon1bd 2961 . . . 4 (𝜑 → (¬ 𝐴𝑉 → [𝐴] = ∅))
2820, 27jcad 513 . . 3 (𝜑 → (¬ 𝐴𝑉 → ((𝐹𝐴) = ∅ ∧ [𝐴] = ∅)))
29 eqtr3 2764 . . 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 1539  wcel 2106  wne 2943  wral 3064  Vcvv 3432  c0 4256  cmpt 5157  dom cdm 5589  cfv 6433   Er wer 8495  [cec 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-er 8498  df-ec 8500
This theorem is referenced by:  ercpbllem  17259  qusaddvallem  17262  qusgrp2  18693  frgpmhm  19371  frgpup3lem  19383  qusring2  19859  qusrhm  20508
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