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Theorem fbasrn 22491
Description: Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
fbasrn.c 𝐶 = ran (𝑥𝐵 ↦ (𝐹𝑥))
Assertion
Ref Expression
fbasrn ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 ∈ (fBas‘𝑌))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝑉   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem fbasrn
Dummy variables 𝑠 𝑟 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fbasrn.c . . 3 𝐶 = ran (𝑥𝐵 ↦ (𝐹𝑥))
2 simpl3 1189 . . . . . 6 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝐵) → 𝑌𝑉)
3 simpl2 1188 . . . . . . 7 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝐵) → 𝐹:𝑋𝑌)
4 fimass 6554 . . . . . . 7 (𝐹:𝑋𝑌 → (𝐹𝑥) ⊆ 𝑌)
53, 4syl 17 . . . . . 6 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝐵) → (𝐹𝑥) ⊆ 𝑌)
62, 5sselpwd 5229 . . . . 5 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ 𝒫 𝑌)
76fmpttd 6878 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵 ↦ (𝐹𝑥)):𝐵⟶𝒫 𝑌)
87frnd 6520 . . 3 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ran (𝑥𝐵 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌)
91, 8eqsstrid 4014 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 ⊆ 𝒫 𝑌)
101a1i 11 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 = ran (𝑥𝐵 ↦ (𝐹𝑥)))
11 ffun 6516 . . . . . . . 8 (𝐹:𝑋𝑌 → Fun 𝐹)
12113ad2ant2 1130 . . . . . . 7 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → Fun 𝐹)
13 funimaexg 6439 . . . . . . . 8 ((Fun 𝐹𝑥𝐵) → (𝐹𝑥) ∈ V)
1413ralrimiva 3182 . . . . . . 7 (Fun 𝐹 → ∀𝑥𝐵 (𝐹𝑥) ∈ V)
15 dmmptg 6095 . . . . . . 7 (∀𝑥𝐵 (𝐹𝑥) ∈ V → dom (𝑥𝐵 ↦ (𝐹𝑥)) = 𝐵)
1612, 14, 153syl 18 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → dom (𝑥𝐵 ↦ (𝐹𝑥)) = 𝐵)
17 fbasne0 22437 . . . . . . 7 (𝐵 ∈ (fBas‘𝑋) → 𝐵 ≠ ∅)
18173ad2ant1 1129 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐵 ≠ ∅)
1916, 18eqnetrd 3083 . . . . 5 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → dom (𝑥𝐵 ↦ (𝐹𝑥)) ≠ ∅)
20 dm0rn0 5794 . . . . . 6 (dom (𝑥𝐵 ↦ (𝐹𝑥)) = ∅ ↔ ran (𝑥𝐵 ↦ (𝐹𝑥)) = ∅)
2120necon3bii 3068 . . . . 5 (dom (𝑥𝐵 ↦ (𝐹𝑥)) ≠ ∅ ↔ ran (𝑥𝐵 ↦ (𝐹𝑥)) ≠ ∅)
2219, 21sylib 220 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ran (𝑥𝐵 ↦ (𝐹𝑥)) ≠ ∅)
2310, 22eqnetrd 3083 . . 3 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 ≠ ∅)
24 fbelss 22440 . . . . . . . . 9 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝑥𝐵) → 𝑥𝑋)
2524ex 415 . . . . . . . 8 (𝐵 ∈ (fBas‘𝑋) → (𝑥𝐵𝑥𝑋))
26253ad2ant1 1129 . . . . . . 7 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵𝑥𝑋))
27 0nelfb 22438 . . . . . . . . . 10 (𝐵 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐵)
28 eleq1 2900 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥𝐵 ↔ ∅ ∈ 𝐵))
2928notbid 320 . . . . . . . . . 10 (𝑥 = ∅ → (¬ 𝑥𝐵 ↔ ¬ ∅ ∈ 𝐵))
3027, 29syl5ibrcom 249 . . . . . . . . 9 (𝐵 ∈ (fBas‘𝑋) → (𝑥 = ∅ → ¬ 𝑥𝐵))
3130con2d 136 . . . . . . . 8 (𝐵 ∈ (fBas‘𝑋) → (𝑥𝐵 → ¬ 𝑥 = ∅))
32313ad2ant1 1129 . . . . . . 7 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵 → ¬ 𝑥 = ∅))
3326, 32jcad 515 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵 → (𝑥𝑋 ∧ ¬ 𝑥 = ∅)))
34 fdm 6521 . . . . . . . . . . . . . . 15 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
35343ad2ant2 1130 . . . . . . . . . . . . . 14 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → dom 𝐹 = 𝑋)
3635sseq2d 3998 . . . . . . . . . . . . 13 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥 ⊆ dom 𝐹𝑥𝑋))
3736biimpar 480 . . . . . . . . . . . 12 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → 𝑥 ⊆ dom 𝐹)
38 sseqin2 4191 . . . . . . . . . . . 12 (𝑥 ⊆ dom 𝐹 ↔ (dom 𝐹𝑥) = 𝑥)
3937, 38sylib 220 . . . . . . . . . . 11 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → (dom 𝐹𝑥) = 𝑥)
4039eqeq1d 2823 . . . . . . . . . 10 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → ((dom 𝐹𝑥) = ∅ ↔ 𝑥 = ∅))
4140biimpd 231 . . . . . . . . 9 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → ((dom 𝐹𝑥) = ∅ → 𝑥 = ∅))
4241con3d 155 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → (¬ 𝑥 = ∅ → ¬ (dom 𝐹𝑥) = ∅))
4342expimpd 456 . . . . . . 7 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝑥𝑋 ∧ ¬ 𝑥 = ∅) → ¬ (dom 𝐹𝑥) = ∅))
44 eqcom 2828 . . . . . . . . 9 (∅ = (𝐹𝑥) ↔ (𝐹𝑥) = ∅)
45 imadisj 5947 . . . . . . . . 9 ((𝐹𝑥) = ∅ ↔ (dom 𝐹𝑥) = ∅)
4644, 45bitri 277 . . . . . . . 8 (∅ = (𝐹𝑥) ↔ (dom 𝐹𝑥) = ∅)
4746notbii 322 . . . . . . 7 (¬ ∅ = (𝐹𝑥) ↔ ¬ (dom 𝐹𝑥) = ∅)
4843, 47syl6ibr 254 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝑥𝑋 ∧ ¬ 𝑥 = ∅) → ¬ ∅ = (𝐹𝑥)))
4933, 48syld 47 . . . . 5 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵 → ¬ ∅ = (𝐹𝑥)))
5049ralrimiv 3181 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ∀𝑥𝐵 ¬ ∅ = (𝐹𝑥))
511eleq2i 2904 . . . . . . 7 (∅ ∈ 𝐶 ↔ ∅ ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)))
52 0ex 5210 . . . . . . . 8 ∅ ∈ V
53 eqid 2821 . . . . . . . . 9 (𝑥𝐵 ↦ (𝐹𝑥)) = (𝑥𝐵 ↦ (𝐹𝑥))
5453elrnmpt 5827 . . . . . . . 8 (∅ ∈ V → (∅ ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐵 ∅ = (𝐹𝑥)))
5552, 54ax-mp 5 . . . . . . 7 (∅ ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐵 ∅ = (𝐹𝑥))
5651, 55bitri 277 . . . . . 6 (∅ ∈ 𝐶 ↔ ∃𝑥𝐵 ∅ = (𝐹𝑥))
5756notbii 322 . . . . 5 (¬ ∅ ∈ 𝐶 ↔ ¬ ∃𝑥𝐵 ∅ = (𝐹𝑥))
58 df-nel 3124 . . . . 5 (∅ ∉ 𝐶 ↔ ¬ ∅ ∈ 𝐶)
59 ralnex 3236 . . . . 5 (∀𝑥𝐵 ¬ ∅ = (𝐹𝑥) ↔ ¬ ∃𝑥𝐵 ∅ = (𝐹𝑥))
6057, 58, 593bitr4i 305 . . . 4 (∅ ∉ 𝐶 ↔ ∀𝑥𝐵 ¬ ∅ = (𝐹𝑥))
6150, 60sylibr 236 . . 3 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ∅ ∉ 𝐶)
621eleq2i 2904 . . . . . . . 8 (𝑟𝐶𝑟 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)))
63 imaeq2 5924 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
6463cbvmptv 5168 . . . . . . . . . 10 (𝑥𝐵 ↦ (𝐹𝑥)) = (𝑢𝐵 ↦ (𝐹𝑢))
6564elrnmpt 5827 . . . . . . . . 9 (𝑟 ∈ V → (𝑟 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑢𝐵 𝑟 = (𝐹𝑢)))
6665elv 3499 . . . . . . . 8 (𝑟 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑢𝐵 𝑟 = (𝐹𝑢))
6762, 66bitri 277 . . . . . . 7 (𝑟𝐶 ↔ ∃𝑢𝐵 𝑟 = (𝐹𝑢))
681eleq2i 2904 . . . . . . . 8 (𝑠𝐶𝑠 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)))
69 imaeq2 5924 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
7069cbvmptv 5168 . . . . . . . . . 10 (𝑥𝐵 ↦ (𝐹𝑥)) = (𝑣𝐵 ↦ (𝐹𝑣))
7170elrnmpt 5827 . . . . . . . . 9 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑣𝐵 𝑠 = (𝐹𝑣)))
7271elv 3499 . . . . . . . 8 (𝑠 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑣𝐵 𝑠 = (𝐹𝑣))
7368, 72bitri 277 . . . . . . 7 (𝑠𝐶 ↔ ∃𝑣𝐵 𝑠 = (𝐹𝑣))
7467, 73anbi12i 628 . . . . . 6 ((𝑟𝐶𝑠𝐶) ↔ (∃𝑢𝐵 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐵 𝑠 = (𝐹𝑣)))
75 reeanv 3367 . . . . . 6 (∃𝑢𝐵𝑣𝐵 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) ↔ (∃𝑢𝐵 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐵 𝑠 = (𝐹𝑣)))
7674, 75bitr4i 280 . . . . 5 ((𝑟𝐶𝑠𝐶) ↔ ∃𝑢𝐵𝑣𝐵 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))
77 fbasssin 22443 . . . . . . . . . . 11 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝑢𝐵𝑣𝐵) → ∃𝑤𝐵 𝑤 ⊆ (𝑢𝑣))
78773expb 1116 . . . . . . . . . 10 ((𝐵 ∈ (fBas‘𝑋) ∧ (𝑢𝐵𝑣𝐵)) → ∃𝑤𝐵 𝑤 ⊆ (𝑢𝑣))
79783ad2antl1 1181 . . . . . . . . 9 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑢𝐵𝑣𝐵)) → ∃𝑤𝐵 𝑤 ⊆ (𝑢𝑣))
8079adantrr 715 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ ((𝑢𝐵𝑣𝐵) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑤𝐵 𝑤 ⊆ (𝑢𝑣))
81 eqid 2821 . . . . . . . . . . . . 13 (𝐹𝑤) = (𝐹𝑤)
82 imaeq2 5924 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
8382rspceeqv 3637 . . . . . . . . . . . . 13 ((𝑤𝐵 ∧ (𝐹𝑤) = (𝐹𝑤)) → ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥))
8481, 83mpan2 689 . . . . . . . . . . . 12 (𝑤𝐵 → ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥))
8584ad2antrl 726 . . . . . . . . . . 11 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥))
861eleq2i 2904 . . . . . . . . . . . . 13 ((𝐹𝑤) ∈ 𝐶 ↔ (𝐹𝑤) ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)))
87 vex 3497 . . . . . . . . . . . . . . 15 𝑤 ∈ V
8887funimaex 6440 . . . . . . . . . . . . . 14 (Fun 𝐹 → (𝐹𝑤) ∈ V)
8953elrnmpt 5827 . . . . . . . . . . . . . 14 ((𝐹𝑤) ∈ V → ((𝐹𝑤) ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥)))
9012, 88, 893syl 18 . . . . . . . . . . . . 13 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝐹𝑤) ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥)))
9186, 90syl5bb 285 . . . . . . . . . . . 12 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝐹𝑤) ∈ 𝐶 ↔ ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥)))
9291ad2antrr 724 . . . . . . . . . . 11 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → ((𝐹𝑤) ∈ 𝐶 ↔ ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥)))
9385, 92mpbird 259 . . . . . . . . . 10 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝐹𝑤) ∈ 𝐶)
94 imass2 5964 . . . . . . . . . . . 12 (𝑤 ⊆ (𝑢𝑣) → (𝐹𝑤) ⊆ (𝐹 “ (𝑢𝑣)))
9594ad2antll 727 . . . . . . . . . . 11 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝐹𝑤) ⊆ (𝐹 “ (𝑢𝑣)))
96 inss1 4204 . . . . . . . . . . . . . 14 (𝑢𝑣) ⊆ 𝑢
97 imass2 5964 . . . . . . . . . . . . . 14 ((𝑢𝑣) ⊆ 𝑢 → (𝐹 “ (𝑢𝑣)) ⊆ (𝐹𝑢))
9896, 97ax-mp 5 . . . . . . . . . . . . 13 (𝐹 “ (𝑢𝑣)) ⊆ (𝐹𝑢)
99 inss2 4205 . . . . . . . . . . . . . 14 (𝑢𝑣) ⊆ 𝑣
100 imass2 5964 . . . . . . . . . . . . . 14 ((𝑢𝑣) ⊆ 𝑣 → (𝐹 “ (𝑢𝑣)) ⊆ (𝐹𝑣))
10199, 100ax-mp 5 . . . . . . . . . . . . 13 (𝐹 “ (𝑢𝑣)) ⊆ (𝐹𝑣)
10298, 101ssini 4207 . . . . . . . . . . . 12 (𝐹 “ (𝑢𝑣)) ⊆ ((𝐹𝑢) ∩ (𝐹𝑣))
103 ineq12 4183 . . . . . . . . . . . . 13 ((𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → (𝑟𝑠) = ((𝐹𝑢) ∩ (𝐹𝑣)))
104103ad2antlr 725 . . . . . . . . . . . 12 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝑟𝑠) = ((𝐹𝑢) ∩ (𝐹𝑣)))
105102, 104sseqtrrid 4019 . . . . . . . . . . 11 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠))
10695, 105sstrd 3976 . . . . . . . . . 10 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝐹𝑤) ⊆ (𝑟𝑠))
107 sseq1 3991 . . . . . . . . . . 11 (𝑧 = (𝐹𝑤) → (𝑧 ⊆ (𝑟𝑠) ↔ (𝐹𝑤) ⊆ (𝑟𝑠)))
108107rspcev 3622 . . . . . . . . . 10 (((𝐹𝑤) ∈ 𝐶 ∧ (𝐹𝑤) ⊆ (𝑟𝑠)) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
10993, 106, 108syl2anc 586 . . . . . . . . 9 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
110109adantlrl 718 . . . . . . . 8 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ ((𝑢𝐵𝑣𝐵) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
11180, 110rexlimddv 3291 . . . . . . 7 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ ((𝑢𝐵𝑣𝐵) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
112111exp32 423 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝑢𝐵𝑣𝐵) → ((𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))))
113112rexlimdvv 3293 . . . . 5 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (∃𝑢𝐵𝑣𝐵 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))
11476, 113syl5bi 244 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝑟𝐶𝑠𝐶) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))
115114ralrimivv 3190 . . 3 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ∀𝑟𝐶𝑠𝐶𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
11623, 61, 1153jca 1124 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟𝐶𝑠𝐶𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))
117 isfbas2 22442 . . 3 (𝑌𝑉 → (𝐶 ∈ (fBas‘𝑌) ↔ (𝐶 ⊆ 𝒫 𝑌 ∧ (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟𝐶𝑠𝐶𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))))
1181173ad2ant3 1131 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝐶 ∈ (fBas‘𝑌) ↔ (𝐶 ⊆ 𝒫 𝑌 ∧ (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟𝐶𝑠𝐶𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))))
1199, 116, 118mpbir2and 711 1 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 ∈ (fBas‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016  wnel 3123  wral 3138  wrex 3139  Vcvv 3494  cin 3934  wss 3935  c0 4290  𝒫 cpw 4538  cmpt 5145  dom cdm 5554  ran crn 5555  cima 5557  Fun wfun 6348  wf 6350  cfv 6354  fBascfbas 20532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-fbas 20541
This theorem is referenced by:  fmfil  22551  fmss  22553  elfm  22554  fmucnd  22900  fmcfil  23874
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