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Theorem fndmdif 6511
Description: Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmdif ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝐴

Proof of Theorem fndmdif
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 difss 3899 . . . . 5 (𝐹𝐺) ⊆ 𝐹
2 dmss 5491 . . . . 5 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
31, 2ax-mp 5 . . . 4 dom (𝐹𝐺) ⊆ dom 𝐹
4 fndm 6168 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54adantr 472 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
63, 5syl5sseq 3813 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) ⊆ 𝐴)
7 sseqin2 3979 . . 3 (dom (𝐹𝐺) ⊆ 𝐴 ↔ (𝐴 ∩ dom (𝐹𝐺)) = dom (𝐹𝐺))
86, 7sylib 209 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹𝐺)) = dom (𝐹𝐺))
9 vex 3353 . . . . 5 𝑥 ∈ V
109eldm 5489 . . . 4 (𝑥 ∈ dom (𝐹𝐺) ↔ ∃𝑦 𝑥(𝐹𝐺)𝑦)
11 eqcom 2772 . . . . . . . . 9 ((𝐹𝑥) = (𝐺𝑥) ↔ (𝐺𝑥) = (𝐹𝑥))
12 fnbrfvb 6424 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐺𝑥) = (𝐹𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1311, 12syl5bb 274 . . . . . . . 8 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (𝐺𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1413adantll 705 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) = (𝐺𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1514necon3abid 2973 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ ¬ 𝑥𝐺(𝐹𝑥)))
16 fvex 6388 . . . . . . 7 (𝐹𝑥) ∈ V
17 breq2 4813 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝑥𝐺𝑦𝑥𝐺(𝐹𝑥)))
1817notbid 309 . . . . . . 7 (𝑦 = (𝐹𝑥) → (¬ 𝑥𝐺𝑦 ↔ ¬ 𝑥𝐺(𝐹𝑥)))
1916, 18ceqsexv 3395 . . . . . 6 (∃𝑦(𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ¬ 𝑥𝐺(𝐹𝑥))
2015, 19syl6bbr 280 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ ∃𝑦(𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦)))
21 eqcom 2772 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
22 fnbrfvb 6424 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
2321, 22syl5bb 274 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
2423adantlr 706 . . . . . . . 8 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
2524anbi1d 623 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦)))
26 brdif 4862 . . . . . . 7 (𝑥(𝐹𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦))
2725, 26syl6bbr 280 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ 𝑥(𝐹𝐺)𝑦))
2827exbidv 2016 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (∃𝑦(𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ∃𝑦 𝑥(𝐹𝐺)𝑦))
2920, 28bitr2d 271 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (∃𝑦 𝑥(𝐹𝐺)𝑦 ↔ (𝐹𝑥) ≠ (𝐺𝑥)))
3010, 29syl5bb 274 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (𝑥 ∈ dom (𝐹𝐺) ↔ (𝐹𝑥) ≠ (𝐺𝑥)))
3130rabbi2dva 3981 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹𝐺)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
328, 31eqtr3d 2801 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  {crab 3059  cdif 3729  cin 3731  wss 3732   class class class wbr 4809  dom cdm 5277   Fn wfn 6063  cfv 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076
This theorem is referenced by:  fndmdifcom  6512  fndmdifeq0  6513  fndifnfp  6635  wemapsolem  8662  wemapso2lem  8664  dsmmbas2  20357  frlmbas  20375  ptcmplem2  22136
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