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Theorem fndmdif 6277
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 3715 . . . . 5 (𝐹𝐺) ⊆ 𝐹
2 dmss 5283 . . . . 5 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
31, 2ax-mp 5 . . . 4 dom (𝐹𝐺) ⊆ dom 𝐹
4 fndm 5948 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54adantr 481 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
63, 5syl5sseq 3632 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) ⊆ 𝐴)
7 sseqin2 3795 . . 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 3189 . . . . 5 𝑥 ∈ V
109eldm 5281 . . . 4 (𝑥 ∈ dom (𝐹𝐺) ↔ ∃𝑦 𝑥(𝐹𝐺)𝑦)
11 eqcom 2628 . . . . . . . . 9 ((𝐹𝑥) = (𝐺𝑥) ↔ (𝐺𝑥) = (𝐹𝑥))
12 fnbrfvb 6193 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐺𝑥) = (𝐹𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1311, 12syl5bb 272 . . . . . . . 8 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (𝐺𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1413adantll 749 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) = (𝐺𝑥) ↔ 𝑥𝐺(𝐹𝑥)))
1514necon3abid 2826 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ ¬ 𝑥𝐺(𝐹𝑥)))
16 fvex 6158 . . . . . . 7 (𝐹𝑥) ∈ V
17 breq2 4617 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝑥𝐺𝑦𝑥𝐺(𝐹𝑥)))
1817notbid 308 . . . . . . 7 (𝑦 = (𝐹𝑥) → (¬ 𝑥𝐺𝑦 ↔ ¬ 𝑥𝐺(𝐹𝑥)))
1916, 18ceqsexv 3228 . . . . . 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 2628 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
22 fnbrfvb 6193 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
2321, 22syl5bb 272 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
2423adantlr 750 . . . . . . . 8 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
2524anbi1d 740 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦)))
26 brdif 4665 . . . . . . 7 (𝑥(𝐹𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦))
2725, 26syl6bbr 278 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ((𝑦 = (𝐹𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ 𝑥(𝐹𝐺)𝑦))
2827exbidv 1847 . . . . 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 3799 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹𝐺)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
328, 31eqtr3d 2657 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 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  {crab 2911  cdif 3552  cin 3554  wss 3555   class class class wbr 4613  dom cdm 5074   Fn wfn 5842  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855
This theorem is referenced by:  fndmdifcom  6278  fndmdifeq0  6279  fndifnfp  6396  wemapsolem  8399  wemapso2lem  8401  dsmmbas2  20000  frlmbas  20018  ptcmplem2  21767
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