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Theorem suppovss 31906
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 3201 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝐶𝐷)
3 suppovss.f . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43fmpo 8054 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹:(𝐴 × 𝐵)⟶𝐷)
52, 4sylib 217 . 2 (𝜑𝐹:(𝐴 × 𝐵)⟶𝐷)
6 simpr 486 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
76fveq2d 6896 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
8 df-ov 7412 . . . . . . . 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 3961 . . . . . . . . 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 836 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶𝐷)
143ovmpt4g 7555 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵𝐶𝐷) → (𝑥𝐹𝑦) = 𝐶)
1510, 11, 13, 14syl3anc 1372 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
168, 15eqtr3id 2787 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
17 suppovss.b . . . . . . . . . . . . . 14 (𝜑𝐵𝑊)
1817adantr 482 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐵𝑊)
1918mptexd 7226 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑦𝐵𝐶) ∈ V)
20 suppovss.g . . . . . . . . . . . 12 𝐺 = (𝑥𝐴 ↦ (𝑦𝐵𝐶))
2119, 20fmptd 7114 . . . . . . . . . . 11 (𝜑𝐺:𝐴⟶V)
22 ssidd 4006 . . . . . . . . . . 11 (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ (𝐺 supp (𝐵 × {𝑍})))
23 suppovss.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
24 snex 5432 . . . . . . . . . . . . 13 {𝑍} ∈ V
2524a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ V)
2617, 25xpexd 7738 . . . . . . . . . . 11 (𝜑 → (𝐵 × {𝑍}) ∈ V)
2721, 22, 23, 26suppssr 8181 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺𝑥) = (𝐵 × {𝑍}))
2827fveq1d 6894 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
2912, 9, 28syl2anc 585 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑥)‘𝑦) = ((𝐵 × {𝑍})‘𝑦))
30 simpr 486 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥𝐴)
3120fvmpt2 7010 . . . . . . . . . . 11 ((𝑥𝐴 ∧ (𝑦𝐵𝐶) ∈ V) → (𝐺𝑥) = (𝑦𝐵𝐶))
3230, 19, 31syl2anc 585 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝑦𝐵𝐶))
331anassrs 469 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝐶𝐷)
3432, 33fvmpt2d 7012 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → ((𝐺𝑥)‘𝑦) = 𝐶)
3512, 10, 11, 34syl21anc 837 . . . . . . . 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 7203 . . . . . . . . 9 ((𝑍𝐷𝑦𝐵) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
3937, 11, 38syl2anc 585 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐵 × {𝑍})‘𝑦) = 𝑍)
4029, 35, 393eqtr3d 2781 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) ∧ 𝑦𝐵) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
417, 16, 403eqtrd 2777 . . . . . 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 5701 . . . . . . 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 483 . . . . 5 ((𝜑𝑧 ∈ ((𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))) × 𝐵)) → ∃𝑥 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
4642, 45r19.29vva 3214 . . . 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 486 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
4948fveq2d 6896 . . . . . . 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 3961 . . . . . . . . 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 836 . . . . . . . . 9 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶𝐷)
5550, 52, 54, 14syl3anc 1372 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥𝐹𝑦) = 𝐶)
568, 55eqtr3id 2787 . . . . . . 7 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
5753, 50, 52, 34syl21anc 837 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑥)‘𝑦) = 𝐶)
58 fvexd 6907 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → ((𝐺𝑥)‘𝑦) ∈ V)
5933, 32, 58fmpt2d 7123 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺𝑥):𝐵⟶V)
60 ssiun2 5051 . . . . . . . . . . . . 13 (𝑥𝐴 → ((𝐺𝑥) supp 𝑍) ⊆ 𝑥𝐴 ((𝐺𝑥) supp 𝑍))
6160adantl 483 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐺𝑥) supp 𝑍) ⊆ 𝑥𝐴 ((𝐺𝑥) supp 𝑍))
62 fveq2 6892 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
6362oveq1d 7424 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → ((𝐺𝑥) supp 𝑍) = ((𝐺𝑘) supp 𝑍))
6463cbviunv 5044 . . . . . . . . . . . 12 𝑥𝐴 ((𝐺𝑥) supp 𝑍) = 𝑘𝐴 ((𝐺𝑘) supp 𝑍)
6561, 64sseqtrdi 4033 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐺𝑥) supp 𝑍) ⊆ 𝑘𝐴 ((𝐺𝑘) supp 𝑍))
66 simpl 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝜑)
67 simpr 486 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
6867eldifad 3961 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → 𝑘𝐴)
6921, 22, 23, 26suppssr 8181 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → (𝐺𝑘) = (𝐵 × {𝑍}))
70 eleq1w 2817 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑘 → (𝑥𝐴𝑘𝐴))
7170anbi2d 630 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → ((𝜑𝑥𝐴) ↔ (𝜑𝑘𝐴)))
7262fneq1d 6643 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → ((𝐺𝑥) Fn 𝐵 ↔ (𝐺𝑘) Fn 𝐵))
7371, 72imbi12d 345 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑘 → (((𝜑𝑥𝐴) → (𝐺𝑥) Fn 𝐵) ↔ ((𝜑𝑘𝐴) → (𝐺𝑘) Fn 𝐵)))
7459ffnd 6719 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐺𝑥) Fn 𝐵)
7573, 74chvarvv 2003 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐺𝑘) Fn 𝐵)
7617adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝐵𝑊)
7736adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝑍𝐷)
78 fnsuppeq0 8177 . . . . . . . . . . . . . . . . . 18 (((𝐺𝑘) Fn 𝐵𝐵𝑊𝑍𝐷) → (((𝐺𝑘) supp 𝑍) = ∅ ↔ (𝐺𝑘) = (𝐵 × {𝑍})))
7975, 76, 77, 78syl3anc 1372 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (((𝐺𝑘) supp 𝑍) = ∅ ↔ (𝐺𝑘) = (𝐵 × {𝑍})))
8079biimpar 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐴) ∧ (𝐺𝑘) = (𝐵 × {𝑍})) → ((𝐺𝑘) supp 𝑍) = ∅)
8166, 68, 69, 80syl21anc 837 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) → ((𝐺𝑘) supp 𝑍) = ∅)
8281ralrimiva 3147 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘 ∈ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))((𝐺𝑘) supp 𝑍) = ∅)
83 nfcv 2904 . . . . . . . . . . . . . . 15 𝑘(𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))
8483iunxdif3 5099 . . . . . . . . . . . . . 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 4268 . . . . . . . . . . . . . . 15 (𝐴 ∩ (𝐺 supp (𝐵 × {𝑍}))) = (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))
87 suppssdm 8162 . . . . . . . . . . . . . . . . 17 (𝐺 supp (𝐵 × {𝑍})) ⊆ dom 𝐺
8887, 21fssdm 6738 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺 supp (𝐵 × {𝑍})) ⊆ 𝐴)
89 sseqin2 4216 . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍})))) = (𝐺 supp (𝐵 × {𝑍})))
9291iuneq1d 5025 . . . . . . . . . . . . 13 (𝜑 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ (𝐺 supp (𝐵 × {𝑍}))))((𝐺𝑘) supp 𝑍) = 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9385, 92eqtr3d 2775 . . . . . . . . . . . 12 (𝜑 𝑘𝐴 ((𝐺𝑘) supp 𝑍) = 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9493adantr 482 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑘𝐴 ((𝐺𝑘) supp 𝑍) = 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9565, 94sseqtrd 4023 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝐺𝑥) supp 𝑍) ⊆ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))
9636adantr 482 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑍𝐷)
9759, 95, 18, 96suppssr 8181 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) → ((𝐺𝑥)‘𝑦) = 𝑍)
9897adantr 482 . . . . . . . 8 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑥)‘𝑦) = 𝑍)
9957, 98eqtr3d 2775 . . . . . . 7 ((((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐵 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍))) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐶 = 𝑍)
10049, 56, 993eqtrd 2777 . . . . . 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 5701 . . . . . . 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 483 . . . . 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 3214 . . . 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 486 . . . . 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 6164 . . . . 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 2844 . . . 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 4149 . . . 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 958 . 2 ((𝜑𝑧 ∈ ((𝐴 × 𝐵) ∖ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))) → (𝐹𝑧) = 𝑍)
1135, 112suppss 8179 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 397  wo 846   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  cdif 3946  cun 3947  cin 3948  wss 3949  c0 4323  {csn 4629  cop 4635   ciun 4998  cmpt 5232   × cxp 5675   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  cmpo 7411   supp csupp 8146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-supp 8147
This theorem is referenced by:  fedgmullem1  32714
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