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Theorem fbasrn 21437
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 simpl2 1057 . . . . . . 7 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝐵) → 𝐹:𝑋𝑌)
3 imassrn 5380 . . . . . . . 8 (𝐹𝑥) ⊆ ran 𝐹
4 frn 5949 . . . . . . . 8 (𝐹:𝑋𝑌 → ran 𝐹𝑌)
53, 4syl5ss 3575 . . . . . . 7 (𝐹:𝑋𝑌 → (𝐹𝑥) ⊆ 𝑌)
62, 5syl 17 . . . . . 6 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝐵) → (𝐹𝑥) ⊆ 𝑌)
7 simpl3 1058 . . . . . . 7 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝐵) → 𝑌𝑉)
8 elpw2g 4746 . . . . . . 7 (𝑌𝑉 → ((𝐹𝑥) ∈ 𝒫 𝑌 ↔ (𝐹𝑥) ⊆ 𝑌))
97, 8syl 17 . . . . . 6 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝐵) → ((𝐹𝑥) ∈ 𝒫 𝑌 ↔ (𝐹𝑥) ⊆ 𝑌))
106, 9mpbird 245 . . . . 5 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ 𝒫 𝑌)
11 eqid 2606 . . . . 5 (𝑥𝐵 ↦ (𝐹𝑥)) = (𝑥𝐵 ↦ (𝐹𝑥))
1210, 11fmptd 6274 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵 ↦ (𝐹𝑥)):𝐵⟶𝒫 𝑌)
13 frn 5949 . . . 4 ((𝑥𝐵 ↦ (𝐹𝑥)):𝐵⟶𝒫 𝑌 → ran (𝑥𝐵 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌)
1412, 13syl 17 . . 3 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ran (𝑥𝐵 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌)
151, 14syl5eqss 3608 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 ⊆ 𝒫 𝑌)
161a1i 11 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 = ran (𝑥𝐵 ↦ (𝐹𝑥)))
17 ffun 5944 . . . . . . . 8 (𝐹:𝑋𝑌 → Fun 𝐹)
18173ad2ant2 1075 . . . . . . 7 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → Fun 𝐹)
19 funimaexg 5872 . . . . . . . 8 ((Fun 𝐹𝑥𝐵) → (𝐹𝑥) ∈ V)
2019ralrimiva 2945 . . . . . . 7 (Fun 𝐹 → ∀𝑥𝐵 (𝐹𝑥) ∈ V)
21 dmmptg 5532 . . . . . . 7 (∀𝑥𝐵 (𝐹𝑥) ∈ V → dom (𝑥𝐵 ↦ (𝐹𝑥)) = 𝐵)
2218, 20, 213syl 18 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → dom (𝑥𝐵 ↦ (𝐹𝑥)) = 𝐵)
23 fbasne0 21383 . . . . . . 7 (𝐵 ∈ (fBas‘𝑋) → 𝐵 ≠ ∅)
24233ad2ant1 1074 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐵 ≠ ∅)
2522, 24eqnetrd 2845 . . . . 5 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → dom (𝑥𝐵 ↦ (𝐹𝑥)) ≠ ∅)
26 dm0rn0 5247 . . . . . 6 (dom (𝑥𝐵 ↦ (𝐹𝑥)) = ∅ ↔ ran (𝑥𝐵 ↦ (𝐹𝑥)) = ∅)
2726necon3bii 2830 . . . . 5 (dom (𝑥𝐵 ↦ (𝐹𝑥)) ≠ ∅ ↔ ran (𝑥𝐵 ↦ (𝐹𝑥)) ≠ ∅)
2825, 27sylib 206 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ran (𝑥𝐵 ↦ (𝐹𝑥)) ≠ ∅)
2916, 28eqnetrd 2845 . . 3 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 ≠ ∅)
30 fbelss 21386 . . . . . . . . 9 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝑥𝐵) → 𝑥𝑋)
3130ex 448 . . . . . . . 8 (𝐵 ∈ (fBas‘𝑋) → (𝑥𝐵𝑥𝑋))
32313ad2ant1 1074 . . . . . . 7 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵𝑥𝑋))
33 0nelfb 21384 . . . . . . . . . 10 (𝐵 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐵)
34 eleq1 2672 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥𝐵 ↔ ∅ ∈ 𝐵))
3534notbid 306 . . . . . . . . . 10 (𝑥 = ∅ → (¬ 𝑥𝐵 ↔ ¬ ∅ ∈ 𝐵))
3633, 35syl5ibrcom 235 . . . . . . . . 9 (𝐵 ∈ (fBas‘𝑋) → (𝑥 = ∅ → ¬ 𝑥𝐵))
3736con2d 127 . . . . . . . 8 (𝐵 ∈ (fBas‘𝑋) → (𝑥𝐵 → ¬ 𝑥 = ∅))
38373ad2ant1 1074 . . . . . . 7 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵 → ¬ 𝑥 = ∅))
3932, 38jcad 553 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵 → (𝑥𝑋 ∧ ¬ 𝑥 = ∅)))
40 fdm 5947 . . . . . . . . . . . . . . 15 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
41403ad2ant2 1075 . . . . . . . . . . . . . 14 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → dom 𝐹 = 𝑋)
4241sseq2d 3592 . . . . . . . . . . . . 13 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥 ⊆ dom 𝐹𝑥𝑋))
4342biimpar 500 . . . . . . . . . . . 12 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → 𝑥 ⊆ dom 𝐹)
44 sseqin2 3775 . . . . . . . . . . . 12 (𝑥 ⊆ dom 𝐹 ↔ (dom 𝐹𝑥) = 𝑥)
4543, 44sylib 206 . . . . . . . . . . 11 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → (dom 𝐹𝑥) = 𝑥)
4645eqeq1d 2608 . . . . . . . . . 10 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → ((dom 𝐹𝑥) = ∅ ↔ 𝑥 = ∅))
4746biimpd 217 . . . . . . . . 9 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → ((dom 𝐹𝑥) = ∅ → 𝑥 = ∅))
4847con3d 146 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ 𝑥𝑋) → (¬ 𝑥 = ∅ → ¬ (dom 𝐹𝑥) = ∅))
4948expimpd 626 . . . . . . 7 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝑥𝑋 ∧ ¬ 𝑥 = ∅) → ¬ (dom 𝐹𝑥) = ∅))
50 eqcom 2613 . . . . . . . . 9 (∅ = (𝐹𝑥) ↔ (𝐹𝑥) = ∅)
51 imadisj 5387 . . . . . . . . 9 ((𝐹𝑥) = ∅ ↔ (dom 𝐹𝑥) = ∅)
5250, 51bitri 262 . . . . . . . 8 (∅ = (𝐹𝑥) ↔ (dom 𝐹𝑥) = ∅)
5352notbii 308 . . . . . . 7 (¬ ∅ = (𝐹𝑥) ↔ ¬ (dom 𝐹𝑥) = ∅)
5449, 53syl6ibr 240 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝑥𝑋 ∧ ¬ 𝑥 = ∅) → ¬ ∅ = (𝐹𝑥)))
5539, 54syld 45 . . . . 5 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝑥𝐵 → ¬ ∅ = (𝐹𝑥)))
5655ralrimiv 2944 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ∀𝑥𝐵 ¬ ∅ = (𝐹𝑥))
571eleq2i 2676 . . . . . . 7 (∅ ∈ 𝐶 ↔ ∅ ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)))
58 0ex 4710 . . . . . . . 8 ∅ ∈ V
5911elrnmpt 5277 . . . . . . . 8 (∅ ∈ V → (∅ ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐵 ∅ = (𝐹𝑥)))
6058, 59ax-mp 5 . . . . . . 7 (∅ ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐵 ∅ = (𝐹𝑥))
6157, 60bitri 262 . . . . . 6 (∅ ∈ 𝐶 ↔ ∃𝑥𝐵 ∅ = (𝐹𝑥))
6261notbii 308 . . . . 5 (¬ ∅ ∈ 𝐶 ↔ ¬ ∃𝑥𝐵 ∅ = (𝐹𝑥))
63 df-nel 2779 . . . . 5 (∅ ∉ 𝐶 ↔ ¬ ∅ ∈ 𝐶)
64 ralnex 2971 . . . . 5 (∀𝑥𝐵 ¬ ∅ = (𝐹𝑥) ↔ ¬ ∃𝑥𝐵 ∅ = (𝐹𝑥))
6562, 63, 643bitr4i 290 . . . 4 (∅ ∉ 𝐶 ↔ ∀𝑥𝐵 ¬ ∅ = (𝐹𝑥))
6656, 65sylibr 222 . . 3 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ∅ ∉ 𝐶)
671eleq2i 2676 . . . . . . . 8 (𝑟𝐶𝑟 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)))
68 vex 3172 . . . . . . . . 9 𝑟 ∈ V
69 imaeq2 5365 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
7069cbvmptv 4669 . . . . . . . . . 10 (𝑥𝐵 ↦ (𝐹𝑥)) = (𝑢𝐵 ↦ (𝐹𝑢))
7170elrnmpt 5277 . . . . . . . . 9 (𝑟 ∈ V → (𝑟 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑢𝐵 𝑟 = (𝐹𝑢)))
7268, 71ax-mp 5 . . . . . . . 8 (𝑟 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑢𝐵 𝑟 = (𝐹𝑢))
7367, 72bitri 262 . . . . . . 7 (𝑟𝐶 ↔ ∃𝑢𝐵 𝑟 = (𝐹𝑢))
741eleq2i 2676 . . . . . . . 8 (𝑠𝐶𝑠 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)))
75 vex 3172 . . . . . . . . 9 𝑠 ∈ V
76 imaeq2 5365 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
7776cbvmptv 4669 . . . . . . . . . 10 (𝑥𝐵 ↦ (𝐹𝑥)) = (𝑣𝐵 ↦ (𝐹𝑣))
7877elrnmpt 5277 . . . . . . . . 9 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑣𝐵 𝑠 = (𝐹𝑣)))
7975, 78ax-mp 5 . . . . . . . 8 (𝑠 ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑣𝐵 𝑠 = (𝐹𝑣))
8074, 79bitri 262 . . . . . . 7 (𝑠𝐶 ↔ ∃𝑣𝐵 𝑠 = (𝐹𝑣))
8173, 80anbi12i 728 . . . . . 6 ((𝑟𝐶𝑠𝐶) ↔ (∃𝑢𝐵 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐵 𝑠 = (𝐹𝑣)))
82 reeanv 3082 . . . . . 6 (∃𝑢𝐵𝑣𝐵 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) ↔ (∃𝑢𝐵 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐵 𝑠 = (𝐹𝑣)))
8381, 82bitr4i 265 . . . . 5 ((𝑟𝐶𝑠𝐶) ↔ ∃𝑢𝐵𝑣𝐵 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))
84 fbasssin 21389 . . . . . . . . . . 11 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝑢𝐵𝑣𝐵) → ∃𝑤𝐵 𝑤 ⊆ (𝑢𝑣))
85843expb 1257 . . . . . . . . . 10 ((𝐵 ∈ (fBas‘𝑋) ∧ (𝑢𝐵𝑣𝐵)) → ∃𝑤𝐵 𝑤 ⊆ (𝑢𝑣))
86853ad2antl1 1215 . . . . . . . . 9 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑢𝐵𝑣𝐵)) → ∃𝑤𝐵 𝑤 ⊆ (𝑢𝑣))
8786adantrr 748 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ ((𝑢𝐵𝑣𝐵) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑤𝐵 𝑤 ⊆ (𝑢𝑣))
88 eqid 2606 . . . . . . . . . . . . 13 (𝐹𝑤) = (𝐹𝑤)
89 imaeq2 5365 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
9089eqeq2d 2616 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → ((𝐹𝑤) = (𝐹𝑥) ↔ (𝐹𝑤) = (𝐹𝑤)))
9190rspcev 3278 . . . . . . . . . . . . 13 ((𝑤𝐵 ∧ (𝐹𝑤) = (𝐹𝑤)) → ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥))
9288, 91mpan2 702 . . . . . . . . . . . 12 (𝑤𝐵 → ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥))
9392ad2antrl 759 . . . . . . . . . . 11 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥))
941eleq2i 2676 . . . . . . . . . . . . 13 ((𝐹𝑤) ∈ 𝐶 ↔ (𝐹𝑤) ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)))
95 vex 3172 . . . . . . . . . . . . . . 15 𝑤 ∈ V
9695funimaex 5873 . . . . . . . . . . . . . 14 (Fun 𝐹 → (𝐹𝑤) ∈ V)
9711elrnmpt 5277 . . . . . . . . . . . . . 14 ((𝐹𝑤) ∈ V → ((𝐹𝑤) ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥)))
9818, 96, 973syl 18 . . . . . . . . . . . . 13 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝐹𝑤) ∈ ran (𝑥𝐵 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥)))
9994, 98syl5bb 270 . . . . . . . . . . . 12 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝐹𝑤) ∈ 𝐶 ↔ ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥)))
10099ad2antrr 757 . . . . . . . . . . 11 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → ((𝐹𝑤) ∈ 𝐶 ↔ ∃𝑥𝐵 (𝐹𝑤) = (𝐹𝑥)))
10193, 100mpbird 245 . . . . . . . . . 10 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝐹𝑤) ∈ 𝐶)
102 imass2 5404 . . . . . . . . . . . 12 (𝑤 ⊆ (𝑢𝑣) → (𝐹𝑤) ⊆ (𝐹 “ (𝑢𝑣)))
103102ad2antll 760 . . . . . . . . . . 11 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝐹𝑤) ⊆ (𝐹 “ (𝑢𝑣)))
104 inss1 3791 . . . . . . . . . . . . . 14 (𝑢𝑣) ⊆ 𝑢
105 imass2 5404 . . . . . . . . . . . . . 14 ((𝑢𝑣) ⊆ 𝑢 → (𝐹 “ (𝑢𝑣)) ⊆ (𝐹𝑢))
106104, 105ax-mp 5 . . . . . . . . . . . . 13 (𝐹 “ (𝑢𝑣)) ⊆ (𝐹𝑢)
107 inss2 3792 . . . . . . . . . . . . . 14 (𝑢𝑣) ⊆ 𝑣
108 imass2 5404 . . . . . . . . . . . . . 14 ((𝑢𝑣) ⊆ 𝑣 → (𝐹 “ (𝑢𝑣)) ⊆ (𝐹𝑣))
109107, 108ax-mp 5 . . . . . . . . . . . . 13 (𝐹 “ (𝑢𝑣)) ⊆ (𝐹𝑣)
110106, 109ssini 3794 . . . . . . . . . . . 12 (𝐹 “ (𝑢𝑣)) ⊆ ((𝐹𝑢) ∩ (𝐹𝑣))
111 ineq12 3767 . . . . . . . . . . . . 13 ((𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → (𝑟𝑠) = ((𝐹𝑢) ∩ (𝐹𝑣)))
112111ad2antlr 758 . . . . . . . . . . . 12 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝑟𝑠) = ((𝐹𝑢) ∩ (𝐹𝑣)))
113110, 112syl5sseqr 3613 . . . . . . . . . . 11 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠))
114103, 113sstrd 3574 . . . . . . . . . 10 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → (𝐹𝑤) ⊆ (𝑟𝑠))
115 sseq1 3585 . . . . . . . . . . 11 (𝑧 = (𝐹𝑤) → (𝑧 ⊆ (𝑟𝑠) ↔ (𝐹𝑤) ⊆ (𝑟𝑠)))
116115rspcev 3278 . . . . . . . . . 10 (((𝐹𝑤) ∈ 𝐶 ∧ (𝐹𝑤) ⊆ (𝑟𝑠)) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
117101, 114, 116syl2anc 690 . . . . . . . . 9 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
118117adantlrl 751 . . . . . . . 8 ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ ((𝑢𝐵𝑣𝐵) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) ∧ (𝑤𝐵𝑤 ⊆ (𝑢𝑣))) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
11987, 118rexlimddv 3013 . . . . . . 7 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) ∧ ((𝑢𝐵𝑣𝐵) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
120119exp32 628 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝑢𝐵𝑣𝐵) → ((𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠))))
121120rexlimdvv 3015 . . . . 5 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (∃𝑢𝐵𝑣𝐵 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))
12283, 121syl5bi 230 . . . 4 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ((𝑟𝐶𝑠𝐶) → ∃𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))
123122ralrimivv 2949 . . 3 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → ∀𝑟𝐶𝑠𝐶𝑧𝐶 𝑧 ⊆ (𝑟𝑠))
12429, 66, 1233jca 1234 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟𝐶𝑠𝐶𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))
125 isfbas2 21388 . . 3 (𝑌𝑉 → (𝐶 ∈ (fBas‘𝑌) ↔ (𝐶 ⊆ 𝒫 𝑌 ∧ (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟𝐶𝑠𝐶𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))))
1261253ad2ant3 1076 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → (𝐶 ∈ (fBas‘𝑌) ↔ (𝐶 ⊆ 𝒫 𝑌 ∧ (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟𝐶𝑠𝐶𝑧𝐶 𝑧 ⊆ (𝑟𝑠)))))
12715, 124, 126mpbir2and 958 1 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 ∈ (fBas‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2776  wnel 2777  wral 2892  wrex 2893  Vcvv 3169  cin 3535  wss 3536  c0 3870  𝒫 cpw 4104  cmpt 4634  dom cdm 5025  ran crn 5026  cima 5028  Fun wfun 5781  wf 5783  cfv 5787  fBascfbas 19498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-fv 5795  df-fbas 19507
This theorem is referenced by:  fmfil  21497  fmss  21499  elfm  21500  fmucnd  21845  fmcfil  22793
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