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Theorem fndmdif 6476
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 3872 . . . . 5 (𝐹𝐺) ⊆ 𝐹
2 dmss 5470 . . . . 5 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
31, 2ax-mp 5 . . . 4 dom (𝐹𝐺) ⊆ dom 𝐹
4 fndm 6143 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54adantr 472 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
63, 5syl5sseq 3786 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) ⊆ 𝐴)
7 sseqin2 3952 . . 3 (dom (𝐹𝐺) ⊆ 𝐴 ↔ (𝐴 ∩ dom (𝐹𝐺)) = dom (𝐹𝐺))
86, 7sylib 208 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹𝐺)) = dom (𝐹𝐺))
9 vex 3335 . . . . 5 𝑥 ∈ V
109eldm 5468 . . . 4 (𝑥 ∈ dom (𝐹𝐺) ↔ ∃𝑦 𝑥(𝐹𝐺)𝑦)
11 eqcom 2759 . . . . . . . . 9 ((𝐹𝑥) = (𝐺𝑥) ↔ (𝐺𝑥) = (𝐹𝑥))
12 fnbrfvb 6389 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐺𝑥) = (𝐹𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1311, 12syl5bb 272 . . . . . . . 8 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (𝐺𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1413adantll 752 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) = (𝐺𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1514necon3abid 2960 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ ¬ 𝑥𝐺(𝐹𝑥)))
16 fvex 6354 . . . . . . 7 (𝐹𝑥) ∈ V
17 breq2 4800 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝑥𝐺𝑦𝑥𝐺(𝐹𝑥)))
1817notbid 307 . . . . . . 7 (𝑦 = (𝐹𝑥) → (¬ 𝑥𝐺𝑦 ↔ ¬ 𝑥𝐺(𝐹𝑥)))
1916, 18ceqsexv 3374 . . . . . 6 (∃𝑦(𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ¬ 𝑥𝐺(𝐹𝑥))
2015, 19syl6bbr 278 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ ∃𝑦(𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦)))
21 eqcom 2759 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
22 fnbrfvb 6389 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
2321, 22syl5bb 272 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
2423adantlr 753 . . . . . . . 8 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
2524anbi1d 743 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦)))
26 brdif 4849 . . . . . . 7 (𝑥(𝐹𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦))
2725, 26syl6bbr 278 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ 𝑥(𝐹𝐺)𝑦))
2827exbidv 1991 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (∃𝑦(𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ∃𝑦 𝑥(𝐹𝐺)𝑦))
2920, 28bitr2d 269 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (∃𝑦 𝑥(𝐹𝐺)𝑦 ↔ (𝐹𝑥) ≠ (𝐺𝑥)))
3010, 29syl5bb 272 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (𝑥 ∈ dom (𝐹𝐺) ↔ (𝐹𝑥) ≠ (𝐺𝑥)))
3130rabbi2dva 3956 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹𝐺)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
328, 31eqtr3d 2788 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1624  wex 1845  wcel 2131  wne 2924  {crab 3046  cdif 3704  cin 3706  wss 3707   class class class wbr 4796  dom cdm 5258   Fn wfn 6036  cfv 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fn 6044  df-fv 6049
This theorem is referenced by:  fndmdifcom  6477  fndmdifeq0  6478  fndifnfp  6598  wemapsolem  8612  wemapso2lem  8614  dsmmbas2  20275  frlmbas  20293  ptcmplem2  22050
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