ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fndmdif GIF version

Theorem fndmdif 5621
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 3261 . . . . 5 (𝐹𝐺) ⊆ 𝐹
2 dmss 4826 . . . . 5 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
31, 2ax-mp 5 . . . 4 dom (𝐹𝐺) ⊆ dom 𝐹
4 fndm 5315 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54adantr 276 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
63, 5sseqtrid 3205 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) ⊆ 𝐴)
7 dfss1 3339 . . 3 (dom (𝐹𝐺) ⊆ 𝐴 ↔ (𝐴 ∩ dom (𝐹𝐺)) = dom (𝐹𝐺))
86, 7sylib 122 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹𝐺)) = dom (𝐹𝐺))
9 vex 2740 . . . . 5 𝑥 ∈ V
109eldm 4824 . . . 4 (𝑥 ∈ dom (𝐹𝐺) ↔ ∃𝑦 𝑥(𝐹𝐺)𝑦)
11 eqcom 2179 . . . . . . . 8 ((𝐹𝑥) = (𝐺𝑥) ↔ (𝐺𝑥) = (𝐹𝑥))
12 fnbrfvb 5556 . . . . . . . 8 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐺𝑥) = (𝐹𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1311, 12bitrid 192 . . . . . . 7 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (𝐺𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1413adantll 476 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) = (𝐺𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1514necon3abid 2386 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ ¬ 𝑥𝐺(𝐹𝑥)))
16 funfvex 5532 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
1716funfni 5316 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
1817adantlr 477 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ V)
19 breq2 4007 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝑥𝐺𝑦𝑥𝐺(𝐹𝑥)))
2019notbid 667 . . . . . . 7 (𝑦 = (𝐹𝑥) → (¬ 𝑥𝐺𝑦 ↔ ¬ 𝑥𝐺(𝐹𝑥)))
2120ceqsexgv 2866 . . . . . 6 ((𝐹𝑥) ∈ V → (∃𝑦(𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ¬ 𝑥𝐺(𝐹𝑥)))
2218, 21syl 14 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (∃𝑦(𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ¬ 𝑥𝐺(𝐹𝑥)))
23 eqcom 2179 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
24 fnbrfvb 5556 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
2523, 24bitrid 192 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
2625adantlr 477 . . . . . . . 8 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
2726anbi1d 465 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦)))
28 brdif 4056 . . . . . . 7 (𝑥(𝐹𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦))
2927, 28bitr4di 198 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ 𝑥(𝐹𝐺)𝑦))
3029exbidv 1825 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (∃𝑦(𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ∃𝑦 𝑥(𝐹𝐺)𝑦))
3115, 22, 303bitr2rd 217 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (∃𝑦 𝑥(𝐹𝐺)𝑦 ↔ (𝐹𝑥) ≠ (𝐺𝑥)))
3210, 31bitrid 192 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (𝑥 ∈ dom (𝐹𝐺) ↔ (𝐹𝑥) ≠ (𝐺𝑥)))
3332rabbi2dva 3343 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹𝐺)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
348, 33eqtr3d 2212 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  wne 2347  {crab 2459  Vcvv 2737  cdif 3126  cin 3128  wss 3129   class class class wbr 4003  dom cdm 4626   Fn wfn 5211  cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224
This theorem is referenced by:  fndmdifcom  5622
  Copyright terms: Public domain W3C validator