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Theorem pstmfval 29718
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 29717 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
323ad2ant1 1080 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
43oveqd 6621 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ))
5 fvex 6158 . . . . . 6 (~Met𝐷) ∈ V
61, 5eqeltri 2694 . . . . 5 ∈ V
76ecelqsi 7748 . . . 4 (𝐴𝑋 → [𝐴] ∈ (𝑋 / ))
873ad2ant2 1081 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → [𝐴] ∈ (𝑋 / ))
96ecelqsi 7748 . . . 4 (𝐵𝑋 → [𝐵] ∈ (𝑋 / ))
1093ad2ant3 1082 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → [𝐵] ∈ (𝑋 / ))
11 rexeq 3128 . . . . . 6 (𝑥 = [𝐴] → (∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)))
1211abbidv 2738 . . . . 5 (𝑥 = [𝐴] → {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
1312unieqd 4412 . . . 4 (𝑥 = [𝐴] {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
14 rexeq 3128 . . . . . . 7 (𝑦 = [𝐵] → (∃𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)))
1514rexbidv 3045 . . . . . 6 (𝑦 = [𝐵] → (∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)))
1615abbidv 2738 . . . . 5 (𝑦 = [𝐵] → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
1716unieqd 4412 . . . 4 (𝑦 = [𝐵] {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
18 eqid 2621 . . . 4 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
19 ecexg 7691 . . . . . . 7 ( ∈ V → [𝐴] ∈ V)
206, 19ax-mp 5 . . . . . 6 [𝐴] ∈ V
21 ecexg 7691 . . . . . . 7 ( ∈ V → [𝐵] ∈ V)
226, 21ax-mp 5 . . . . . 6 [𝐵] ∈ V
2320, 22ab2rexex 7104 . . . . 5 {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} ∈ V
2423uniex 6906 . . . 4 {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} ∈ V
2513, 17, 18, 24ovmpt2 6749 . . 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 692 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ) = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
27 simpr3 1067 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝑒𝐷𝑓))
28 simpl1 1062 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐷 ∈ (PsMet‘𝑋))
29 simpr1 1065 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑒 ∈ [𝐴] )
30 metidss 29713 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
311, 30syl5eqss 3628 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ (PsMet‘𝑋) → ⊆ (𝑋 × 𝑋))
32 xpss 5187 . . . . . . . . . . . . . . . . . . 19 (𝑋 × 𝑋) ⊆ (V × V)
3331, 32syl6ss 3595 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ (PsMet‘𝑋) → ⊆ (V × V))
34 df-rel 5081 . . . . . . . . . . . . . . . . . 18 (Rel ⊆ (V × V))
3533, 34sylibr 224 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (PsMet‘𝑋) → Rel )
36353ad2ant1 1080 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → Rel )
3736adantr 481 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → Rel )
38 relelec 7732 . . . . . . . . . . . . . . 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 4619 . . . . . . . . . . . . 13 (𝐴 𝑒𝐴(~Met𝐷)𝑒)
4240, 41sylib 208 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐴(~Met𝐷)𝑒)
43 simpr2 1066 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑓 ∈ [𝐵] )
44 relelec 7732 . . . . . . . . . . . . . . 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 4619 . . . . . . . . . . . . 13 (𝐵 𝑓𝐵(~Met𝐷)𝑓)
4846, 47sylib 208 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐵(~Met𝐷)𝑓)
49 metideq 29715 . . . . . . . . . . . 12 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝑒𝐵(~Met𝐷)𝑓)) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
5028, 42, 48, 49syl12anc 1321 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
5127, 50eqtr4d 2658 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
5251adantlr 750 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
53523anassrs 1287 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) ∧ 𝑒 ∈ [𝐴] ) ∧ 𝑓 ∈ [𝐵] ) ∧ 𝑧 = (𝑒𝐷𝑓)) → 𝑧 = (𝐴𝐷𝐵))
54 oveq1 6611 . . . . . . . . . . . 12 (𝑎 = 𝑒 → (𝑎𝐷𝑏) = (𝑒𝐷𝑏))
5554eqeq2d 2631 . . . . . . . . . . 11 (𝑎 = 𝑒 → (𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑏)))
56 oveq2 6612 . . . . . . . . . . . 12 (𝑏 = 𝑓 → (𝑒𝐷𝑏) = (𝑒𝐷𝑓))
5756eqeq2d 2631 . . . . . . . . . . 11 (𝑏 = 𝑓 → (𝑧 = (𝑒𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑓)))
5855, 57cbvrex2v 3168 . . . . . . . . . 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 482 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) → ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
6153, 60r19.29vva 3073 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) → 𝑧 = (𝐴𝐷𝐵))
62 simpl1 1062 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐷 ∈ (PsMet‘𝑋))
63 simpl2 1063 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴𝑋)
64 psmet0 22023 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)
6562, 63, 64syl2anc 692 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴𝐷𝐴) = 0)
66 relelec 7732 . . . . . . . . . . 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 4624 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 𝐴𝐴(~Met𝐷)𝐴))
70 metidv 29714 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐴𝑋)) → (𝐴(~Met𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7162, 63, 63, 70syl12anc 1321 . . . . . . . . . 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 1064 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵𝑋)
75 psmet0 22023 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵𝑋) → (𝐵𝐷𝐵) = 0)
7662, 74, 75syl2anc 692 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵𝐷𝐵) = 0)
77 relelec 7732 . . . . . . . . . . 11 (Rel → (𝐵 ∈ [𝐵] 𝐵 𝐵))
7862, 35, 773syl 18 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] 𝐵 𝐵))
7968breqd 4624 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 𝐵𝐵(~Met𝐷)𝐵))
80 metidv 29714 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐵𝑋)) → (𝐵(~Met𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
8162, 74, 74, 80syl12anc 1321 . . . . . . . . . 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 477 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝑧 = (𝐴𝐷𝐵))
85 rspceov 6645 . . . . . . . 8 ((𝐴 ∈ [𝐴] 𝐵 ∈ [𝐵] 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏))
8673, 83, 84, 85syl3anc 1323 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏))
8761, 86impbida 876 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝐴𝐷𝐵)))
8887abbidv 2738 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {𝑧𝑧 = (𝐴𝐷𝐵)})
89 df-sn 4149 . . . . 5 {(𝐴𝐷𝐵)} = {𝑧𝑧 = (𝐴𝐷𝐵)}
9088, 89syl6eqr 2673 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
9190unieqd 4412 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
92 ovex 6632 . . . 4 (𝐴𝐷𝐵) ∈ V
9392unisn 4417 . . 3 {(𝐴𝐷𝐵)} = (𝐴𝐷𝐵)
9491, 93syl6eq 2671 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = (𝐴𝐷𝐵))
954, 26, 943eqtrd 2659 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  Vcvv 3186  wss 3555  {csn 4148   cuni 4402   class class class wbr 4613   × cxp 5072  Rel wrel 5079  cfv 5847  (class class class)co 6604  cmpt2 6606  [cec 7685   / cqs 7686  0cc0 9880  PsMetcpsmet 19649  ~Metcmetid 29708  pstoMetcpstm 29709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-xadd 11891  df-psmet 19657  df-metid 29710  df-pstm 29711
This theorem is referenced by:  pstmxmet  29719
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