MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnelfmlem Structured version   Visualization version   GIF version

Theorem rnelfmlem 21737
Description: Lemma for rnelfm 21738. (Contributed by Jeff Hankins, 14-Nov-2009.)
Assertion
Ref Expression
rnelfmlem (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐿   𝑥,𝑋   𝑥,𝑌

Proof of Theorem rnelfmlem
Dummy variables 𝑟 𝑠 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1064 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐹:𝑌𝑋)
2 cnvimass 5473 . . . . . . . 8 (𝐹𝑥) ⊆ dom 𝐹
3 fdm 6038 . . . . . . . 8 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
42, 3syl5sseq 3645 . . . . . . 7 (𝐹:𝑌𝑋 → (𝐹𝑥) ⊆ 𝑌)
51, 4syl 17 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑥) ⊆ 𝑌)
6 simpl1 1062 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌𝐴)
7 elpw2g 4818 . . . . . . 7 (𝑌𝐴 → ((𝐹𝑥) ∈ 𝒫 𝑌 ↔ (𝐹𝑥) ⊆ 𝑌))
86, 7syl 17 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝐹𝑥) ∈ 𝒫 𝑌 ↔ (𝐹𝑥) ⊆ 𝑌))
95, 8mpbird 247 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑥) ∈ 𝒫 𝑌)
109adantr 481 . . . 4 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝐹𝑥) ∈ 𝒫 𝑌)
11 eqid 2620 . . . 4 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
1210, 11fmptd 6371 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑥𝐿 ↦ (𝐹𝑥)):𝐿⟶𝒫 𝑌)
13 frn 6040 . . 3 ((𝑥𝐿 ↦ (𝐹𝑥)):𝐿⟶𝒫 𝑌 → ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌)
1412, 13syl 17 . 2 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌)
15 filtop 21640 . . . . . . . 8 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
16153ad2ant2 1081 . . . . . . 7 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑋𝐿)
1716adantr 481 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑋𝐿)
18 fimacnv 6333 . . . . . . . . 9 (𝐹:𝑌𝑋 → (𝐹𝑋) = 𝑌)
1918eqcomd 2626 . . . . . . . 8 (𝐹:𝑌𝑋𝑌 = (𝐹𝑋))
20193ad2ant3 1082 . . . . . . 7 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑌 = (𝐹𝑋))
2120adantr 481 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌 = (𝐹𝑋))
22 imaeq2 5450 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2322eqeq2d 2630 . . . . . . 7 (𝑥 = 𝑋 → (𝑌 = (𝐹𝑥) ↔ 𝑌 = (𝐹𝑋)))
2423rspcev 3304 . . . . . 6 ((𝑋𝐿𝑌 = (𝐹𝑋)) → ∃𝑥𝐿 𝑌 = (𝐹𝑥))
2517, 21, 24syl2anc 692 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ∃𝑥𝐿 𝑌 = (𝐹𝑥))
2611elrnmpt 5361 . . . . . . 7 (𝑌𝐴 → (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑌 = (𝐹𝑥)))
27263ad2ant1 1080 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑌 = (𝐹𝑥)))
2827adantr 481 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑌 = (𝐹𝑥)))
2925, 28mpbird 247 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
30 ne0i 3913 . . . 4 (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅)
3129, 30syl 17 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅)
32 0nelfil 21634 . . . . . . 7 (𝐿 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐿)
33323ad2ant2 1081 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → ¬ ∅ ∈ 𝐿)
3433adantr 481 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ¬ ∅ ∈ 𝐿)
35 0ex 4781 . . . . . . 7 ∅ ∈ V
3611elrnmpt 5361 . . . . . . 7 (∅ ∈ V → (∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 ∅ = (𝐹𝑥)))
3735, 36ax-mp 5 . . . . . 6 (∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 ∅ = (𝐹𝑥))
38 ffn 6032 . . . . . . . . . . . . . . . . . 18 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
39 fvelrnb 6230 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝑌 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
4038, 39syl 17 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌𝑋 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
41403ad2ant3 1082 . . . . . . . . . . . . . . . 16 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
4241ad2antrr 761 . . . . . . . . . . . . . . 15 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
43 eleq1 2687 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑥𝑦𝑥))
4443biimparc 504 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑥 ∧ (𝐹𝑧) = 𝑦) → (𝐹𝑧) ∈ 𝑥)
4544ad2ant2l 781 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝐿𝑦𝑥) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → (𝐹𝑧) ∈ 𝑥)
4645adantll 749 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → (𝐹𝑧) ∈ 𝑥)
47 ffun 6035 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:𝑌𝑋 → Fun 𝐹)
48473ad2ant3 1082 . . . . . . . . . . . . . . . . . . . 20 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → Fun 𝐹)
4948ad3antrrr 765 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → Fun 𝐹)
503eleq2d 2685 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:𝑌𝑋 → (𝑧 ∈ dom 𝐹𝑧𝑌))
5150biimpar 502 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:𝑌𝑋𝑧𝑌) → 𝑧 ∈ dom 𝐹)
52513ad2antl3 1223 . . . . . . . . . . . . . . . . . . . . 21 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ 𝑧𝑌) → 𝑧 ∈ dom 𝐹)
5352adantlr 750 . . . . . . . . . . . . . . . . . . . 20 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑧𝑌) → 𝑧 ∈ dom 𝐹)
5453ad2ant2r 782 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → 𝑧 ∈ dom 𝐹)
55 fvimacnv 6318 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
5649, 54, 55syl2anc 692 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
5746, 56mpbid 222 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → 𝑧 ∈ (𝐹𝑥))
58 n0i 3912 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐹𝑥) → ¬ (𝐹𝑥) = ∅)
59 eqcom 2627 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) = ∅ ↔ ∅ = (𝐹𝑥))
6058, 59sylnib 318 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐹𝑥) → ¬ ∅ = (𝐹𝑥))
6157, 60syl 17 . . . . . . . . . . . . . . . 16 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → ¬ ∅ = (𝐹𝑥))
6261rexlimdvaa 3028 . . . . . . . . . . . . . . 15 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (∃𝑧𝑌 (𝐹𝑧) = 𝑦 → ¬ ∅ = (𝐹𝑥)))
6342, 62sylbid 230 . . . . . . . . . . . . . 14 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (𝑦 ∈ ran 𝐹 → ¬ ∅ = (𝐹𝑥)))
6463con2d 129 . . . . . . . . . . . . 13 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (∅ = (𝐹𝑥) → ¬ 𝑦 ∈ ran 𝐹))
6564expr 642 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝑦𝑥 → (∅ = (𝐹𝑥) → ¬ 𝑦 ∈ ran 𝐹)))
6665com23 86 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (∅ = (𝐹𝑥) → (𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹)))
6766impr 648 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → (𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹))
6867alrimiv 1853 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ∀𝑦(𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹))
69 imnan 438 . . . . . . . . . . . 12 ((𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ¬ (𝑦𝑥𝑦 ∈ ran 𝐹))
70 elin 3788 . . . . . . . . . . . 12 (𝑦 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑦𝑥𝑦 ∈ ran 𝐹))
7169, 70xchbinxr 325 . . . . . . . . . . 11 ((𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹))
7271albii 1745 . . . . . . . . . 10 (∀𝑦(𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ∀𝑦 ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹))
73 eq0 3921 . . . . . . . . . 10 ((𝑥 ∩ ran 𝐹) = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹))
74 eqcom 2627 . . . . . . . . . 10 ((𝑥 ∩ ran 𝐹) = ∅ ↔ ∅ = (𝑥 ∩ ran 𝐹))
7572, 73, 743bitr2i 288 . . . . . . . . 9 (∀𝑦(𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ∅ = (𝑥 ∩ ran 𝐹))
7668, 75sylib 208 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ∅ = (𝑥 ∩ ran 𝐹))
77 simpll2 1099 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → 𝐿 ∈ (Fil‘𝑋))
78 simprl 793 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → 𝑥𝐿)
79 simplr 791 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ran 𝐹𝐿)
80 filin 21639 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
8177, 78, 79, 80syl3anc 1324 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
8276, 81eqeltrd 2699 . . . . . . 7 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ∅ ∈ 𝐿)
8382rexlimdvaa 3028 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑥𝐿 ∅ = (𝐹𝑥) → ∅ ∈ 𝐿))
8437, 83syl5bi 232 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ∅ ∈ 𝐿))
8534, 84mtod 189 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ¬ ∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
86 df-nel 2895 . . . 4 (∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ¬ ∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
8785, 86sylibr 224 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)))
88 vex 3198 . . . . . . . . 9 𝑟 ∈ V
8911elrnmpt 5361 . . . . . . . . 9 (𝑟 ∈ V → (𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑟 = (𝐹𝑥)))
9088, 89ax-mp 5 . . . . . . . 8 (𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑟 = (𝐹𝑥))
91 imaeq2 5450 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
9291eqeq2d 2630 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑟 = (𝐹𝑥) ↔ 𝑟 = (𝐹𝑢)))
9392cbvrexv 3167 . . . . . . . 8 (∃𝑥𝐿 𝑟 = (𝐹𝑥) ↔ ∃𝑢𝐿 𝑟 = (𝐹𝑢))
9490, 93bitri 264 . . . . . . 7 (𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑢𝐿 𝑟 = (𝐹𝑢))
95 vex 3198 . . . . . . . . 9 𝑠 ∈ V
9611elrnmpt 5361 . . . . . . . . 9 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
9795, 96ax-mp 5 . . . . . . . 8 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
98 imaeq2 5450 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
9998eqeq2d 2630 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑠 = (𝐹𝑥) ↔ 𝑠 = (𝐹𝑣)))
10099cbvrexv 3167 . . . . . . . 8 (∃𝑥𝐿 𝑠 = (𝐹𝑥) ↔ ∃𝑣𝐿 𝑠 = (𝐹𝑣))
10197, 100bitri 264 . . . . . . 7 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑣𝐿 𝑠 = (𝐹𝑣))
10294, 101anbi12i 732 . . . . . 6 ((𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (∃𝑢𝐿 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐿 𝑠 = (𝐹𝑣)))
103 reeanv 3102 . . . . . 6 (∃𝑢𝐿𝑣𝐿 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) ↔ (∃𝑢𝐿 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐿 𝑠 = (𝐹𝑣)))
104102, 103bitr4i 267 . . . . 5 ((𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ ∃𝑢𝐿𝑣𝐿 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))
105 filin 21639 . . . . . . . . . . . . . 14 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑢𝐿𝑣𝐿) → (𝑢𝑣) ∈ 𝐿)
1061053expb 1264 . . . . . . . . . . . . 13 ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑢𝐿𝑣𝐿)) → (𝑢𝑣) ∈ 𝐿)
107106adantlr 750 . . . . . . . . . . . 12 (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → (𝑢𝑣) ∈ 𝐿)
108 eqidd 2621 . . . . . . . . . . . 12 (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → (𝐹 “ (𝑢𝑣)) = (𝐹 “ (𝑢𝑣)))
109 imaeq2 5450 . . . . . . . . . . . . . 14 (𝑥 = (𝑢𝑣) → (𝐹𝑥) = (𝐹 “ (𝑢𝑣)))
110109eqeq2d 2630 . . . . . . . . . . . . 13 (𝑥 = (𝑢𝑣) → ((𝐹 “ (𝑢𝑣)) = (𝐹𝑥) ↔ (𝐹 “ (𝑢𝑣)) = (𝐹 “ (𝑢𝑣))))
111110rspcev 3304 . . . . . . . . . . . 12 (((𝑢𝑣) ∈ 𝐿 ∧ (𝐹 “ (𝑢𝑣)) = (𝐹 “ (𝑢𝑣))) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
112107, 108, 111syl2anc 692 . . . . . . . . . . 11 (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
1131123adantl1 1215 . . . . . . . . . 10 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
114113ad2ant2r 782 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
115 simpll1 1098 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → 𝑌𝐴)
116 cnvimass 5473 . . . . . . . . . . . . . 14 (𝐹 “ (𝑢𝑣)) ⊆ dom 𝐹
117116, 3syl5sseq 3645 . . . . . . . . . . . . 13 (𝐹:𝑌𝑋 → (𝐹 “ (𝑢𝑣)) ⊆ 𝑌)
1181173ad2ant3 1082 . . . . . . . . . . . 12 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹 “ (𝑢𝑣)) ⊆ 𝑌)
119118ad2antrr 761 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ⊆ 𝑌)
120115, 119ssexd 4796 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ∈ V)
12111elrnmpt 5361 . . . . . . . . . 10 ((𝐹 “ (𝑢𝑣)) ∈ V → ((𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥)))
122120, 121syl 17 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ((𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥)))
123114, 122mpbird 247 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
124 simprrl 803 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → 𝑟 = (𝐹𝑢))
125 simprrr 804 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → 𝑠 = (𝐹𝑣))
126124, 125ineq12d 3807 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝑟𝑠) = ((𝐹𝑢) ∩ (𝐹𝑣)))
127 funcnvcnv 5944 . . . . . . . . . . . . 13 (Fun 𝐹 → Fun 𝐹)
128 imain 5962 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
12947, 127, 1283syl 18 . . . . . . . . . . . 12 (𝐹:𝑌𝑋 → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
1301293ad2ant3 1082 . . . . . . . . . . 11 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
131130ad2antrr 761 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
132126, 131eqtr4d 2657 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝑟𝑠) = (𝐹 “ (𝑢𝑣)))
133 eqimss2 3650 . . . . . . . . 9 ((𝑟𝑠) = (𝐹 “ (𝑢𝑣)) → (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠))
134132, 133syl 17 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠))
135 sseq1 3618 . . . . . . . . 9 (𝑡 = (𝐹 “ (𝑢𝑣)) → (𝑡 ⊆ (𝑟𝑠) ↔ (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠)))
136135rspcev 3304 . . . . . . . 8 (((𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠)) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))
137123, 134, 136syl2anc 692 . . . . . . 7 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))
138137exp32 630 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑢𝐿𝑣𝐿) → ((𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))))
139138rexlimdvv 3033 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑢𝐿𝑣𝐿 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))
140104, 139syl5bi 232 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))
141140ralrimivv 2967 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))
14231, 87, 1413jca 1240 . 2 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅ ∧ ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))
143 isfbas2 21620 . . 3 (𝑌𝐴 → (ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌) ↔ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌 ∧ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅ ∧ ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))))
1446, 143syl 17 . 2 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌) ↔ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌 ∧ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅ ∧ ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))))
14514, 142, 144mpbir2and 956 1 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wal 1479   = wceq 1481  wcel 1988  wne 2791  wnel 2894  wral 2909  wrex 2910  Vcvv 3195  cin 3566  wss 3567  c0 3907  𝒫 cpw 4149  cmpt 4720  ccnv 5103  dom cdm 5104  ran crn 5105  cima 5107  Fun wfun 5870   Fn wfn 5871  wf 5872  cfv 5876  fBascfbas 19715  Filcfil 21630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-fbas 19724  df-fil 21631
This theorem is referenced by:  rnelfm  21738  fmfnfm  21743
  Copyright terms: Public domain W3C validator