Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  suppovss Structured version   Visualization version   GIF version

Theorem suppovss 31441
Description: A bound for the support of an operation. (Contributed by Thierry Arnoux, 19-Jul-2023.)
Hypotheses
Ref Expression
suppovss.f 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
suppovss.g 𝐺 = (𝑥𝐴 ↦ (𝑦𝐵𝐶))
suppovss.a (𝜑𝐴𝑉)
suppovss.b (𝜑𝐵𝑊)
suppovss.z (𝜑𝑍𝐷)
suppovss.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝐷)
Assertion
Ref Expression
suppovss (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))
Distinct variable groups:   𝐴,𝑘,𝑥,𝑦   𝐵,𝑘,𝑥,𝑦   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦   𝑘,𝐺,𝑥,𝑦   𝑘,𝑍,𝑥,𝑦   𝜑,𝑘,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑘)   𝐷(𝑘)   𝐹(𝑘)   𝑉(𝑥,𝑦,𝑘)   𝑊(𝑥,𝑦,𝑘)

Proof of Theorem suppovss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 suppovss.1 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝐷)
21ralrimivva 3195 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝐶𝐷)
3 suppovss.f . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43fmpo 7992 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹:(𝐴 × 𝐵)⟶𝐷)
52, 4sylib 217 . 2 (𝜑𝐹:(𝐴 × 𝐵)⟶𝐷)
6 simpr 485 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
76fveq2d 6843 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
8 df-ov 7354 . . . . . . . 8 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
9 simpllr 774 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
109eldifad 3920 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥𝐴)
11 simplr 767 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦𝐵)
12 simplll 773 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝜑)
1312, 10, 11, 1syl12anc 835 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶𝐷)
143ovmpt4g 7496 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵𝐶𝐷) → (𝑥𝐹𝑦) = 𝐶)
1510, 11, 13, 14syl3anc 1371 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
168, 15eqtr3id 2790 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
17 suppovss.b . . . . . . . . . . . . . 14 (𝜑𝐵𝑊)
1817adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐵𝑊)
1918mptexd 7170 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑦𝐵𝐶) ∈ V)
20 suppovss.g . . . . . . . . . . . 12 𝐺 = (𝑥𝐴 ↦ (𝑦𝐵𝐶))
2119, 20fmptd 7058 . . . . . . . . . . 11 (𝜑𝐺:𝐴⟶V)
22 ssidd 3965 . . . . . . . . . . 11 (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ (𝐺 supp (𝐵 × {𝑍})))
23 suppovss.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
24 snex 5386 . . . . . . . . . . . . 13 {𝑍} ∈ V
2524a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ V)
2617, 25xpexd 7677 . . . . . . . . . . 11 (𝜑 → (𝐵 × {𝑍}) ∈ V)
2721, 22, 23, 26suppssr 8119 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺𝑥) = (𝐵 × {𝑍}))
2827fveq1d 6841 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
2912, 9, 28syl2anc 584 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
30 simpr 485 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥𝐴)
3120fvmpt2 6956 . . . . . . . . . . 11 ((𝑥𝐴 ∧ (𝑦𝐵𝐶) ∈ V) → (𝐺𝑥) = (𝑦𝐵𝐶))
3230, 19, 31syl2anc 584 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝑦𝐵𝐶))
331anassrs 468 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝐶𝐷)
3432, 33fvmpt2d 6958 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → ((𝐺𝑥)‘𝑦) = 𝐶)
3512, 10, 11, 34syl21anc 836 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑥)‘𝑦) = 𝐶)
36 suppovss.z . . . . . . . . . 10 (𝜑𝑍𝐷)
3712, 36syl 17 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑍𝐷)
38 fvconst2g 7147 . . . . . . . . 9 ((𝑍𝐷𝑦𝐵) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
3937, 11, 38syl2anc 584 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
4029, 35, 393eqtr3d 2784 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
417, 16, 403eqtrd 2780 . . . . . 6 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = 𝑍)
4241adantl3r 748 . . . . 5 (((((𝜑𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = 𝑍)
43 elxp2 5655 . . . . . . 7 (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
4443biimpi 215 . . . . . 6 (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) → ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
4544adantl 482 . . . . 5 ((𝜑𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
4642, 45r19.29vva 3205 . . . 4 ((𝜑𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → (𝐹𝑧) = 𝑍)
4746adantlr 713 . . 3 (((𝜑𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) ∧ 𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → (𝐹𝑧) = 𝑍)
48 simpr 485 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
4948fveq2d 6843 . . . . . . 7 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
50 simpllr 774 . . . . . . . . 9 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥𝐴)
51 simplr 767 . . . . . . . . . 10 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))
5251eldifad 3920 . . . . . . . . 9 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦𝐵)
53 simplll 773 . . . . . . . . . 10 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝜑)
5453, 50, 52, 1syl12anc 835 . . . . . . . . 9 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶𝐷)
5550, 52, 54, 14syl3anc 1371 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
568, 55eqtr3id 2790 . . . . . . 7 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
5753, 50, 52, 34syl21anc 836 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑥)‘𝑦) = 𝐶)
58 fvexd 6854 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → ((𝐺𝑥)‘𝑦) ∈ V)
5933, 32, 58fmpt2d 7067 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥):𝐵⟶V)
60 ssiun2 5005 . . . . . . . . . . . . 13 (𝑥𝐴 → ((𝐺𝑥) supp 𝑍) ⊆ 𝑥𝐴 ((𝐺𝑥) supp 𝑍))
6160adantl 482 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐺𝑥) supp 𝑍) ⊆ 𝑥𝐴 ((𝐺𝑥) supp 𝑍))
62 fveq2 6839 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
6362oveq1d 7366 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → ((𝐺𝑥) supp 𝑍) = ((𝐺𝑘) supp 𝑍))
6463cbviunv 4998 . . . . . . . . . . . 12 𝑥𝐴 ((𝐺𝑥) supp 𝑍) = 𝑘𝐴 ((𝐺𝑘) supp 𝑍)
6561, 64sseqtrdi 3992 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐺𝑥) supp 𝑍) ⊆ 𝑘𝐴 ((𝐺𝑘) supp 𝑍))
66 simpl 483 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝜑)
67 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
6867eldifad 3920 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘𝐴)
6921, 22, 23, 26suppssr 8119 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺𝑘) = (𝐵 × {𝑍}))
70 eleq1w 2820 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑘 → (𝑥𝐴𝑘𝐴))
7170anbi2d 629 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → ((𝜑𝑥𝐴) ↔ (𝜑𝑘𝐴)))
7262fneq1d 6592 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → ((𝐺𝑥) Fn 𝐵 ↔ (𝐺𝑘) Fn 𝐵))
7371, 72imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑘 → (((𝜑𝑥𝐴) → (𝐺𝑥) Fn 𝐵) ↔ ((𝜑𝑘𝐴) → (𝐺𝑘) Fn 𝐵)))
7459ffnd 6666 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐺𝑥) Fn 𝐵)
7573, 74chvarvv 2002 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺𝑘) Fn 𝐵)
7617adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝐵𝑊)
7736adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝑍𝐷)
78 fnsuppeq0 8115 . . . . . . . . . . . . . . . . . 18 (((𝐺𝑘) Fn 𝐵𝐵𝑊𝑍𝐷) → (((𝐺𝑘) supp 𝑍) = ∅ ↔ (𝐺𝑘) = (𝐵 × {𝑍})))
7975, 76, 77, 78syl3anc 1371 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (((𝐺𝑘) supp 𝑍) = ∅ ↔ (𝐺𝑘) = (𝐵 × {𝑍})))
8079biimpar 478 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐴) ∧ (𝐺𝑘) = (𝐵 × {𝑍})) → ((𝐺𝑘) supp 𝑍) = ∅)
8166, 68, 69, 80syl21anc 836 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺𝑘) supp 𝑍) = ∅)
8281ralrimiva 3141 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))((𝐺𝑘) supp 𝑍) = ∅)
83 nfcv 2905 . . . . . . . . . . . . . . 15 𝑘(𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))
8483iunxdif3 5053 . . . . . . . . . . . . . 14 (∀𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))((𝐺𝑘) supp 𝑍) = ∅ → 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺𝑘) supp 𝑍) = 𝑘𝐴 ((𝐺𝑘) supp 𝑍))
8582, 84syl 17 . . . . . . . . . . . . 13 (𝜑 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺𝑘) supp 𝑍) = 𝑘𝐴 ((𝐺𝑘) supp 𝑍))
86 dfin4 4225 . . . . . . . . . . . . . . 15 (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
87 suppssdm 8100 . . . . . . . . . . . . . . . . 17 (𝐺 supp (𝐵 × {𝑍})) ⊆ dom 𝐺
8887, 21fssdm 6685 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴)
89 sseqin2 4173 . . . . . . . . . . . . . . . 16 ((𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
9088, 89sylib 217 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
9186, 90eqtr3id 2790 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) = (𝐺 supp (𝐵 × {𝑍})))
9291iuneq1d 4979 . . . . . . . . . . . . 13 (𝜑 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺𝑘) supp 𝑍) = 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9385, 92eqtr3d 2778 . . . . . . . . . . . 12 (𝜑 𝑘𝐴 ((𝐺𝑘) supp 𝑍) = 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9493adantr 481 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑘𝐴 ((𝐺𝑘) supp 𝑍) = 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9565, 94sseqtrd 3982 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝐺𝑥) supp 𝑍) ⊆ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9636adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑍𝐷)
9759, 95, 18, 96suppssr 8119 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) → ((𝐺𝑥)‘𝑦) = 𝑍)
9897adantr 481 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑥)‘𝑦) = 𝑍)
9957, 98eqtr3d 2778 . . . . . . 7 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
10049, 56, 993eqtrd 2780 . . . . . 6 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = 𝑍)
101100adantl3r 748 . . . . 5 (((((𝜑𝑧 ∈ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = 𝑍)
102 elxp2 5655 . . . . . . 7 (𝑧 ∈ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ↔ ∃𝑥𝐴𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))𝑧 = ⟨𝑥, 𝑦⟩)
103102biimpi 215 . . . . . 6 (𝑧 ∈ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) → ∃𝑥𝐴𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))𝑧 = ⟨𝑥, 𝑦⟩)
104103adantl 482 . . . . 5 ((𝜑𝑧 ∈ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → ∃𝑥𝐴𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))𝑧 = ⟨𝑥, 𝑦⟩)
105101, 104r19.29vva 3205 . . . 4 ((𝜑𝑧 ∈ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → (𝐹𝑧) = 𝑍)
106105adantlr 713 . . 3 (((𝜑𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) ∧ 𝑧 ∈ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → (𝐹𝑧) = 𝑍)
107 simpr 485 . . . . 5 ((𝜑𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → 𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))))
108 difxp 6114 . . . . 5 ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) = (((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∪ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))))
109107, 108eleqtrdi 2848 . . . 4 ((𝜑𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → 𝑧 ∈ (((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∪ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))))
110 elun 4106 . . . 4 (𝑧 ∈ (((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∪ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) ↔ (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∨ 𝑧 ∈ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))))
111109, 110sylib 217 . . 3 ((𝜑𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ∨ 𝑧 ∈ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))))
11247, 106, 111mpjaodan 957 . 2 ((𝜑𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → (𝐹𝑧) = 𝑍)
1135, 112suppss 8117 1 (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3062  wrex 3071  Vcvv 3443  cdif 3905  cun 3906  cin 3907  wss 3908  c0 4280  {csn 4584  cop 4590   ciun 4952  cmpt 5186   × cxp 5629   Fn wfn 6488  wf 6489  cfv 6493  (class class class)co 7351  cmpo 7353   supp csupp 8084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-supp 8085
This theorem is referenced by:  fedgmullem1  32160
  Copyright terms: Public domain W3C validator