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Theorem suppovss 30446
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 3159 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝐶𝐷)
3 suppovss.f . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43fmpo 7752 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹:(𝐴 × 𝐵)⟶𝐷)
52, 4sylib 221 . 2 (𝜑𝐹:(𝐴 × 𝐵)⟶𝐷)
6 simpr 488 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
76fveq2d 6653 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
8 df-ov 7142 . . . . . . . 8 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
9 simpllr 775 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
109eldifad 3896 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥𝐴)
11 simplr 768 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦𝐵)
12 simplll 774 . . . . . . . . . 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 7280 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵𝐶𝐷) → (𝑥𝐹𝑦) = 𝐶)
1510, 11, 13, 14syl3anc 1368 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
168, 15syl5eqr 2850 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
17 suppovss.b . . . . . . . . . . . . . 14 (𝜑𝐵𝑊)
1817adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐵𝑊)
1918mptexd 6968 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑦𝐵𝐶) ∈ V)
20 suppovss.g . . . . . . . . . . . 12 𝐺 = (𝑥𝐴 ↦ (𝑦𝐵𝐶))
2119, 20fmptd 6859 . . . . . . . . . . 11 (𝜑𝐺:𝐴⟶V)
22 ssidd 3941 . . . . . . . . . . 11 (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ (𝐺 supp (𝐵 × {𝑍})))
23 suppovss.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
24 snex 5300 . . . . . . . . . . . . 13 {𝑍} ∈ V
2524a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ V)
2617, 25xpexd 7458 . . . . . . . . . . 11 (𝜑 → (𝐵 × {𝑍}) ∈ V)
2721, 22, 23, 26suppssr 7848 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺𝑥) = (𝐵 × {𝑍}))
2827fveq1d 6651 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
2912, 9, 28syl2anc 587 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
30 simpr 488 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥𝐴)
3120fvmpt2 6760 . . . . . . . . . . 11 ((𝑥𝐴 ∧ (𝑦𝐵𝐶) ∈ V) → (𝐺𝑥) = (𝑦𝐵𝐶))
3230, 19, 31syl2anc 587 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝑦𝐵𝐶))
331anassrs 471 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝐶𝐷)
3432, 33fvmpt2d 6762 . . . . . . . . 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 6945 . . . . . . . . 9 ((𝑍𝐷𝑦𝐵) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
3937, 11, 38syl2anc 587 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
4029, 35, 393eqtr3d 2844 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
417, 16, 403eqtrd 2840 . . . . . 6 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = 𝑍)
4241adantl3r 749 . . . . 5 (((((𝜑𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) ∧ 𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = 𝑍)
43 elxp2 5547 . . . . . . 7 (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
4443biimpi 219 . . . . . 6 (𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵) → ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
4544adantl 485 . . . . 5 ((𝜑𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
4642, 45r19.29vva 3295 . . . 4 ((𝜑𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → (𝐹𝑧) = 𝑍)
4746adantlr 714 . . 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 488 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
4948fveq2d 6653 . . . . . . 7 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
50 simpllr 775 . . . . . . . . 9 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥𝐴)
51 simplr 768 . . . . . . . . . 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 3896 . . . . . . . . 9 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦𝐵)
53 simplll 774 . . . . . . . . . 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 1368 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
568, 55syl5eqr 2850 . . . . . . 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 6664 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → ((𝐺𝑥)‘𝑦) ∈ V)
5933, 32, 58fmpt2d 6868 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥):𝐵⟶V)
60 ssiun2 4937 . . . . . . . . . . . . 13 (𝑥𝐴 → ((𝐺𝑥) supp 𝑍) ⊆ 𝑥𝐴 ((𝐺𝑥) supp 𝑍))
6160adantl 485 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐺𝑥) supp 𝑍) ⊆ 𝑥𝐴 ((𝐺𝑥) supp 𝑍))
62 fveq2 6649 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
6362oveq1d 7154 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → ((𝐺𝑥) supp 𝑍) = ((𝐺𝑘) supp 𝑍))
6463cbviunv 4930 . . . . . . . . . . . 12 𝑥𝐴 ((𝐺𝑥) supp 𝑍) = 𝑘𝐴 ((𝐺𝑘) supp 𝑍)
6561, 64sseqtrdi 3968 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐺𝑥) supp 𝑍) ⊆ 𝑘𝐴 ((𝐺𝑘) supp 𝑍))
66 simpl 486 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝜑)
67 simpr 488 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
6867eldifad 3896 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘𝐴)
6921, 22, 23, 26suppssr 7848 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺𝑘) = (𝐵 × {𝑍}))
70 eleq1w 2875 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑘 → (𝑥𝐴𝑘𝐴))
7170anbi2d 631 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → ((𝜑𝑥𝐴) ↔ (𝜑𝑘𝐴)))
7262fneq1d 6420 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → ((𝐺𝑥) Fn 𝐵 ↔ (𝐺𝑘) Fn 𝐵))
7371, 72imbi12d 348 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑘 → (((𝜑𝑥𝐴) → (𝐺𝑥) Fn 𝐵) ↔ ((𝜑𝑘𝐴) → (𝐺𝑘) Fn 𝐵)))
7459ffnd 6492 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐺𝑥) Fn 𝐵)
7573, 74chvarvv 2005 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺𝑘) Fn 𝐵)
7617adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝐵𝑊)
7736adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝑍𝐷)
78 fnsuppeq0 7845 . . . . . . . . . . . . . . . . . 18 (((𝐺𝑘) Fn 𝐵𝐵𝑊𝑍𝐷) → (((𝐺𝑘) supp 𝑍) = ∅ ↔ (𝐺𝑘) = (𝐵 × {𝑍})))
7975, 76, 77, 78syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (((𝐺𝑘) supp 𝑍) = ∅ ↔ (𝐺𝑘) = (𝐵 × {𝑍})))
8079biimpar 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐴) ∧ (𝐺𝑘) = (𝐵 × {𝑍})) → ((𝐺𝑘) supp 𝑍) = ∅)
8166, 68, 69, 80syl21anc 836 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺𝑘) supp 𝑍) = ∅)
8281ralrimiva 3152 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))((𝐺𝑘) supp 𝑍) = ∅)
83 nfcv 2958 . . . . . . . . . . . . . . 15 𝑘(𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))
8483iunxdif3 4983 . . . . . . . . . . . . . 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 4197 . . . . . . . . . . . . . . 15 (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
87 suppssdm 7830 . . . . . . . . . . . . . . . . 17 (𝐺 supp (𝐵 × {𝑍})) ⊆ dom 𝐺
8887, 21fssdm 6508 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴)
89 sseqin2 4145 . . . . . . . . . . . . . . . 16 ((𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
9088, 89sylib 221 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐺 supp (𝐵 × {𝑍})))
9186, 90syl5eqr 2850 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) = (𝐺 supp (𝐵 × {𝑍})))
9291iuneq1d 4911 . . . . . . . . . . . . 13 (𝜑 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺𝑘) supp 𝑍) = 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9385, 92eqtr3d 2838 . . . . . . . . . . . 12 (𝜑 𝑘𝐴 ((𝐺𝑘) supp 𝑍) = 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9493adantr 484 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑘𝐴 ((𝐺𝑘) supp 𝑍) = 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9565, 94sseqtrd 3958 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝐺𝑥) supp 𝑍) ⊆ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9636adantr 484 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑍𝐷)
9759, 95, 18, 96suppssr 7848 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) → ((𝐺𝑥)‘𝑦) = 𝑍)
9897adantr 484 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑥)‘𝑦) = 𝑍)
9957, 98eqtr3d 2838 . . . . . . 7 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
10049, 56, 993eqtrd 2840 . . . . . 6 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = 𝑍)
101100adantl3r 749 . . . . 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 5547 . . . . . . 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 219 . . . . . 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 485 . . . . 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 3295 . . . 4 ((𝜑𝑧 ∈ (𝐴 × (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → (𝐹𝑧) = 𝑍)
106105adantlr 714 . . 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 488 . . . . 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 5992 . . . . 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 2903 . . . 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 4079 . . . 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 221 . . 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 956 . 2 ((𝜑𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → (𝐹𝑧) = 𝑍)
1135, 112suppss 7847 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 209  wa 399  wo 844   = wceq 1538  wcel 2112  wral 3109  wrex 3110  Vcvv 3444  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  {csn 4528  cop 4534   ciun 4884  cmpt 5113   × cxp 5521   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141   supp csupp 7817
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-supp 7818
This theorem is referenced by:  fedgmullem1  31113
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