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Theorem divsfval 17430
Description: Value of the function in qusval 17425. (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 8695 . . . . 5 (𝜑 → [𝐴] 𝑉)
41, 3ssexd 5282 . . . 4 (𝜑 → [𝐴] ∈ V)
5 eceq1 8687 . . . . 5 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
6 ercpbl.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
75, 6fvmptg 6947 . . . 4 ((𝐴𝑉 ∧ [𝐴] ∈ V) → (𝐹𝐴) = [𝐴] )
84, 7sylan2 594 . . 3 ((𝐴𝑉𝜑) → (𝐹𝐴) = [𝐴] )
98expcom 415 . 2 (𝜑 → (𝐴𝑉 → (𝐹𝐴) = [𝐴] ))
106dmeqi 5861 . . . . . . . 8 dom 𝐹 = dom (𝑥𝑉 ↦ [𝑥] )
112ecss 8695 . . . . . . . . . . 11 (𝜑 → [𝑥] 𝑉)
121, 11ssexd 5282 . . . . . . . . . 10 (𝜑 → [𝑥] ∈ V)
1312ralrimivw 3148 . . . . . . . . 9 (𝜑 → ∀𝑥𝑉 [𝑥] ∈ V)
14 dmmptg 6195 . . . . . . . . 9 (∀𝑥𝑉 [𝑥] ∈ V → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1513, 14syl 17 . . . . . . . 8 (𝜑 → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1610, 15eqtrid 2789 . . . . . . 7 (𝜑 → dom 𝐹 = 𝑉)
1716eleq2d 2824 . . . . . 6 (𝜑 → (𝐴 ∈ dom 𝐹𝐴𝑉))
1817notbid 318 . . . . 5 (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴𝑉))
19 ndmfv 6878 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
2018, 19syl6bir 254 . . . 4 (𝜑 → (¬ 𝐴𝑉 → (𝐹𝐴) = ∅))
21 ecdmn0 8696 . . . . . 6 (𝐴 ∈ dom ↔ [𝐴] ≠ ∅)
22 erdm 8659 . . . . . . . . 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, 25biimtrrid 242 . . . . 5 (𝜑 → ([𝐴] ≠ ∅ → 𝐴𝑉))
2726necon1bd 2962 . . . 4 (𝜑 → (¬ 𝐴𝑉 → [𝐴] = ∅))
2820, 27jcad 514 . . 3 (𝜑 → (¬ 𝐴𝑉 → ((𝐹𝐴) = ∅ ∧ [𝐴] = ∅)))
29 eqtr3 2763 . . 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 397   = wceq 1542  wcel 2107  wne 2944  wral 3065  Vcvv 3446  c0 4283  cmpt 5189  dom cdm 5634  cfv 6497   Er wer 8646  [cec 8647
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-er 8649  df-ec 8651
This theorem is referenced by:  ercpbllem  17431  qusaddvallem  17434  qusgrp2  18866  frgpmhm  19548  frgpup3lem  19560  qusring2  20047  qusrhm  20710
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