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Theorem pstmfval 30248
Description: Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
pstmval.1 = (~Met𝐷)
Assertion
Ref Expression
pstmfval ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = (𝐴𝐷𝐵))

Proof of Theorem pstmfval
Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pstmval.1 . . . . 5 = (~Met𝐷)
21pstmval 30247 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
323ad2ant1 1128 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
43oveqd 6830 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ))
5 fvex 6362 . . . . . 6 (~Met𝐷) ∈ V
61, 5eqeltri 2835 . . . . 5 ∈ V
76ecelqsi 7970 . . . 4 (𝐴𝑋 → [𝐴] ∈ (𝑋 / ))
873ad2ant2 1129 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → [𝐴] ∈ (𝑋 / ))
96ecelqsi 7970 . . . 4 (𝐵𝑋 → [𝐵] ∈ (𝑋 / ))
1093ad2ant3 1130 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → [𝐵] ∈ (𝑋 / ))
11 rexeq 3278 . . . . . 6 (𝑥 = [𝐴] → (∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)))
1211abbidv 2879 . . . . 5 (𝑥 = [𝐴] → {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
1312unieqd 4598 . . . 4 (𝑥 = [𝐴] {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
14 rexeq 3278 . . . . . . 7 (𝑦 = [𝐵] → (∃𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)))
1514rexbidv 3190 . . . . . 6 (𝑦 = [𝐵] → (∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)))
1615abbidv 2879 . . . . 5 (𝑦 = [𝐵] → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
1716unieqd 4598 . . . 4 (𝑦 = [𝐵] {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
18 eqid 2760 . . . 4 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
19 ecexg 7915 . . . . . . 7 ( ∈ V → [𝐴] ∈ V)
206, 19ax-mp 5 . . . . . 6 [𝐴] ∈ V
21 ecexg 7915 . . . . . . 7 ( ∈ V → [𝐵] ∈ V)
226, 21ax-mp 5 . . . . . 6 [𝐵] ∈ V
2320, 22ab2rexex 7324 . . . . 5 {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} ∈ V
2423uniex 7118 . . . 4 {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} ∈ V
2513, 17, 18, 24ovmpt2 6961 . . 3 (([𝐴] ∈ (𝑋 / ) ∧ [𝐵] ∈ (𝑋 / )) → ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ) = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
268, 10, 25syl2anc 696 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ) = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
27 simpr3 1238 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝑒𝐷𝑓))
28 simpl1 1228 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐷 ∈ (PsMet‘𝑋))
29 simpr1 1234 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑒 ∈ [𝐴] )
30 metidss 30243 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
311, 30syl5eqss 3790 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ (PsMet‘𝑋) → ⊆ (𝑋 × 𝑋))
32 xpss 5282 . . . . . . . . . . . . . . . . . . 19 (𝑋 × 𝑋) ⊆ (V × V)
3331, 32syl6ss 3756 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ (PsMet‘𝑋) → ⊆ (V × V))
34 df-rel 5273 . . . . . . . . . . . . . . . . . 18 (Rel ⊆ (V × V))
3533, 34sylibr 224 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (PsMet‘𝑋) → Rel )
36353ad2ant1 1128 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → Rel )
3736adantr 472 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → Rel )
38 relelec 7954 . . . . . . . . . . . . . . 15 (Rel → (𝑒 ∈ [𝐴] 𝐴 𝑒))
3937, 38syl 17 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → (𝑒 ∈ [𝐴] 𝐴 𝑒))
4029, 39mpbid 222 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐴 𝑒)
411breqi 4810 . . . . . . . . . . . . 13 (𝐴 𝑒𝐴(~Met𝐷)𝑒)
4240, 41sylib 208 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐴(~Met𝐷)𝑒)
43 simpr2 1236 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑓 ∈ [𝐵] )
44 relelec 7954 . . . . . . . . . . . . . . 15 (Rel → (𝑓 ∈ [𝐵] 𝐵 𝑓))
4537, 44syl 17 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → (𝑓 ∈ [𝐵] 𝐵 𝑓))
4643, 45mpbid 222 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐵 𝑓)
471breqi 4810 . . . . . . . . . . . . 13 (𝐵 𝑓𝐵(~Met𝐷)𝑓)
4846, 47sylib 208 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐵(~Met𝐷)𝑓)
49 metideq 30245 . . . . . . . . . . . 12 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝑒𝐵(~Met𝐷)𝑓)) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
5028, 42, 48, 49syl12anc 1475 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
5127, 50eqtr4d 2797 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
5251adantlr 753 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
53523anassrs 1454 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) ∧ 𝑒 ∈ [𝐴] ) ∧ 𝑓 ∈ [𝐵] ) ∧ 𝑧 = (𝑒𝐷𝑓)) → 𝑧 = (𝐴𝐷𝐵))
54 oveq1 6820 . . . . . . . . . . . 12 (𝑎 = 𝑒 → (𝑎𝐷𝑏) = (𝑒𝐷𝑏))
5554eqeq2d 2770 . . . . . . . . . . 11 (𝑎 = 𝑒 → (𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑏)))
56 oveq2 6821 . . . . . . . . . . . 12 (𝑏 = 𝑓 → (𝑒𝐷𝑏) = (𝑒𝐷𝑓))
5756eqeq2d 2770 . . . . . . . . . . 11 (𝑏 = 𝑓 → (𝑧 = (𝑒𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑓)))
5855, 57cbvrex2v 3319 . . . . . . . . . 10 (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
5958biimpi 206 . . . . . . . . 9 (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) → ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
6059adantl 473 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) → ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
6153, 60r19.29vva 3219 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) → 𝑧 = (𝐴𝐷𝐵))
62 simpl1 1228 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐷 ∈ (PsMet‘𝑋))
63 simpl2 1230 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴𝑋)
64 psmet0 22314 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)
6562, 63, 64syl2anc 696 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴𝐷𝐴) = 0)
66 relelec 7954 . . . . . . . . . . 11 (Rel → (𝐴 ∈ [𝐴] 𝐴 𝐴))
6762, 35, 663syl 18 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
681a1i 11 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → = (~Met𝐷))
6968breqd 4815 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 𝐴𝐴(~Met𝐷)𝐴))
70 metidv 30244 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐴𝑋)) → (𝐴(~Met𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7162, 63, 63, 70syl12anc 1475 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴(~Met𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7267, 69, 713bitrd 294 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∈ [𝐴] ↔ (𝐴𝐷𝐴) = 0))
7365, 72mpbird 247 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴 ∈ [𝐴] )
74 simpl3 1232 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵𝑋)
75 psmet0 22314 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵𝑋) → (𝐵𝐷𝐵) = 0)
7662, 74, 75syl2anc 696 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵𝐷𝐵) = 0)
77 relelec 7954 . . . . . . . . . . 11 (Rel → (𝐵 ∈ [𝐵] 𝐵 𝐵))
7862, 35, 773syl 18 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] 𝐵 𝐵))
7968breqd 4815 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 𝐵𝐵(~Met𝐷)𝐵))
80 metidv 30244 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐵𝑋)) → (𝐵(~Met𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
8162, 74, 74, 80syl12anc 1475 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵(~Met𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
8278, 79, 813bitrd 294 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] ↔ (𝐵𝐷𝐵) = 0))
8376, 82mpbird 247 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵 ∈ [𝐵] )
84 simpr 479 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝑧 = (𝐴𝐷𝐵))
85 rspceov 6855 . . . . . . . 8 ((𝐴 ∈ [𝐴] 𝐵 ∈ [𝐵] 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏))
8673, 83, 84, 85syl3anc 1477 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏))
8761, 86impbida 913 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝐴𝐷𝐵)))
8887abbidv 2879 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {𝑧𝑧 = (𝐴𝐷𝐵)})
89 df-sn 4322 . . . . 5 {(𝐴𝐷𝐵)} = {𝑧𝑧 = (𝐴𝐷𝐵)}
9088, 89syl6eqr 2812 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
9190unieqd 4598 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
92 ovex 6841 . . . 4 (𝐴𝐷𝐵) ∈ V
9392unisn 4603 . . 3 {(𝐴𝐷𝐵)} = (𝐴𝐷𝐵)
9491, 93syl6eq 2810 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = (𝐴𝐷𝐵))
954, 26, 943eqtrd 2798 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340  wss 3715  {csn 4321   cuni 4588   class class class wbr 4804   × cxp 5264  Rel wrel 5271  cfv 6049  (class class class)co 6813  cmpt2 6815  [cec 7909   / cqs 7910  0cc0 10128  PsMetcpsmet 19932  ~Metcmetid 30238  pstoMetcpstm 30239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-er 7911  df-ec 7913  df-qs 7917  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-xadd 12140  df-psmet 19940  df-metid 30240  df-pstm 30241
This theorem is referenced by:  pstmxmet  30249
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