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Theorem divsfval 16415
Description: Value of the function in qusval 16410. (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 7940 . . . . 5 (𝜑 → [𝐴] 𝑉)
41, 3ssexd 4939 . . . 4 (𝜑 → [𝐴] ∈ V)
5 eceq1 7934 . . . . 5 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
6 ercpbl.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
75, 6fvmptg 6422 . . . 4 ((𝐴𝑉 ∧ [𝐴] ∈ V) → (𝐹𝐴) = [𝐴] )
84, 7sylan2 580 . . 3 ((𝐴𝑉𝜑) → (𝐹𝐴) = [𝐴] )
98expcom 398 . 2 (𝜑 → (𝐴𝑉 → (𝐹𝐴) = [𝐴] ))
106dmeqi 5463 . . . . . . . 8 dom 𝐹 = dom (𝑥𝑉 ↦ [𝑥] )
112ecss 7940 . . . . . . . . . . 11 (𝜑 → [𝑥] 𝑉)
121, 11ssexd 4939 . . . . . . . . . 10 (𝜑 → [𝑥] ∈ V)
1312ralrimivw 3116 . . . . . . . . 9 (𝜑 → ∀𝑥𝑉 [𝑥] ∈ V)
14 dmmptg 5776 . . . . . . . . 9 (∀𝑥𝑉 [𝑥] ∈ V → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1513, 14syl 17 . . . . . . . 8 (𝜑 → dom (𝑥𝑉 ↦ [𝑥] ) = 𝑉)
1610, 15syl5eq 2817 . . . . . . 7 (𝜑 → dom 𝐹 = 𝑉)
1716eleq2d 2836 . . . . . 6 (𝜑 → (𝐴 ∈ dom 𝐹𝐴𝑉))
1817notbid 307 . . . . 5 (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴𝑉))
19 ndmfv 6359 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
2018, 19syl6bir 244 . . . 4 (𝜑 → (¬ 𝐴𝑉 → (𝐹𝐴) = ∅))
21 ecdmn0 7941 . . . . . 6 (𝐴 ∈ dom ↔ [𝐴] ≠ ∅)
22 erdm 7906 . . . . . . . . 9 ( Er 𝑉 → dom = 𝑉)
232, 22syl 17 . . . . . . . 8 (𝜑 → dom = 𝑉)
2423eleq2d 2836 . . . . . . 7 (𝜑 → (𝐴 ∈ dom 𝐴𝑉))
2524biimpd 219 . . . . . 6 (𝜑 → (𝐴 ∈ dom 𝐴𝑉))
2621, 25syl5bir 233 . . . . 5 (𝜑 → ([𝐴] ≠ ∅ → 𝐴𝑉))
2726necon1bd 2961 . . . 4 (𝜑 → (¬ 𝐴𝑉 → [𝐴] = ∅))
2820, 27jcad 502 . . 3 (𝜑 → (¬ 𝐴𝑉 → ((𝐹𝐴) = ∅ ∧ [𝐴] = ∅)))
29 eqtr3 2792 . . 3 (((𝐹𝐴) = ∅ ∧ [𝐴] = ∅) → (𝐹𝐴) = [𝐴] )
3028, 29syl6 35 . 2 (𝜑 → (¬ 𝐴𝑉 → (𝐹𝐴) = [𝐴] ))
319, 30pm2.61d 171 1 (𝜑 → (𝐹𝐴) = [𝐴] )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  Vcvv 3351  c0 4063  cmpt 4863  dom cdm 5249  cfv 6031   Er wer 7893  [cec 7894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-er 7896  df-ec 7898
This theorem is referenced by:  ercpbllem  16416  qusaddvallem  16419  qusgrp2  17741  frgpmhm  18385  frgpup3lem  18397  qusring2  18828  qusrhm  19452
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