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Theorem pstmfval 32477
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 32476 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
323ad2ant1 1133 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
43oveqd 7374 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ))
51fvexi 6856 . . . . 5 ∈ V
65ecelqsi 8712 . . . 4 (𝐴𝑋 → [𝐴] ∈ (𝑋 / ))
763ad2ant2 1134 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → [𝐴] ∈ (𝑋 / ))
85ecelqsi 8712 . . . 4 (𝐵𝑋 → [𝐵] ∈ (𝑋 / ))
983ad2ant3 1135 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → [𝐵] ∈ (𝑋 / ))
10 rexeq 3310 . . . . . 6 (𝑥 = [𝐴] → (∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)))
1110abbidv 2805 . . . . 5 (𝑥 = [𝐴] → {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
1211unieqd 4879 . . . 4 (𝑥 = [𝐴] {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
13 rexeq 3310 . . . . . . 7 (𝑦 = [𝐵] → (∃𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)))
1413rexbidv 3175 . . . . . 6 (𝑦 = [𝐵] → (∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)))
1514abbidv 2805 . . . . 5 (𝑦 = [𝐵] → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
1615unieqd 4879 . . . 4 (𝑦 = [𝐵] {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
17 eqid 2736 . . . 4 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
18 ecexg 8652 . . . . . . 7 ( ∈ V → [𝐴] ∈ V)
195, 18ax-mp 5 . . . . . 6 [𝐴] ∈ V
20 ecexg 8652 . . . . . . 7 ( ∈ V → [𝐵] ∈ V)
215, 20ax-mp 5 . . . . . 6 [𝐵] ∈ V
2219, 21ab2rexex 7912 . . . . 5 {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} ∈ V
2322uniex 7678 . . . 4 {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} ∈ V
2412, 16, 17, 23ovmpo 7515 . . 3 (([𝐴] ∈ (𝑋 / ) ∧ [𝐵] ∈ (𝑋 / )) → ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ) = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
257, 9, 24syl2anc 584 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ) = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
26 simpr3 1196 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝑒𝐷𝑓))
27 simpl1 1191 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐷 ∈ (PsMet‘𝑋))
28 simpr1 1194 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑒 ∈ [𝐴] )
29 metidss 32472 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
301, 29eqsstrid 3992 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ (PsMet‘𝑋) → ⊆ (𝑋 × 𝑋))
31 xpss 5649 . . . . . . . . . . . . . . . . . . 19 (𝑋 × 𝑋) ⊆ (V × V)
3230, 31sstrdi 3956 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ (PsMet‘𝑋) → ⊆ (V × V))
33 df-rel 5640 . . . . . . . . . . . . . . . . . 18 (Rel ⊆ (V × V))
3432, 33sylibr 233 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (PsMet‘𝑋) → Rel )
35343ad2ant1 1133 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → Rel )
3635adantr 481 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → Rel )
37 relelec 8693 . . . . . . . . . . . . . . 15 (Rel → (𝑒 ∈ [𝐴] 𝐴 𝑒))
3836, 37syl 17 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → (𝑒 ∈ [𝐴] 𝐴 𝑒))
3928, 38mpbid 231 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐴 𝑒)
401breqi 5111 . . . . . . . . . . . . 13 (𝐴 𝑒𝐴(~Met𝐷)𝑒)
4139, 40sylib 217 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐴(~Met𝐷)𝑒)
42 simpr2 1195 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑓 ∈ [𝐵] )
43 relelec 8693 . . . . . . . . . . . . . . 15 (Rel → (𝑓 ∈ [𝐵] 𝐵 𝑓))
4436, 43syl 17 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → (𝑓 ∈ [𝐵] 𝐵 𝑓))
4542, 44mpbid 231 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐵 𝑓)
461breqi 5111 . . . . . . . . . . . . 13 (𝐵 𝑓𝐵(~Met𝐷)𝑓)
4745, 46sylib 217 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐵(~Met𝐷)𝑓)
48 metideq 32474 . . . . . . . . . . . 12 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝑒𝐵(~Met𝐷)𝑓)) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
4927, 41, 47, 48syl12anc 835 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
5026, 49eqtr4d 2779 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
5150adantlr 713 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
52513anassrs 1360 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) ∧ 𝑒 ∈ [𝐴] ) ∧ 𝑓 ∈ [𝐵] ) ∧ 𝑧 = (𝑒𝐷𝑓)) → 𝑧 = (𝐴𝐷𝐵))
53 oveq1 7364 . . . . . . . . . . . 12 (𝑎 = 𝑒 → (𝑎𝐷𝑏) = (𝑒𝐷𝑏))
5453eqeq2d 2747 . . . . . . . . . . 11 (𝑎 = 𝑒 → (𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑏)))
55 oveq2 7365 . . . . . . . . . . . 12 (𝑏 = 𝑓 → (𝑒𝐷𝑏) = (𝑒𝐷𝑓))
5655eqeq2d 2747 . . . . . . . . . . 11 (𝑏 = 𝑓 → (𝑧 = (𝑒𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑓)))
5754, 56cbvrex2vw 3228 . . . . . . . . . 10 (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
5857biimpi 215 . . . . . . . . 9 (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) → ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
5958adantl 482 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) → ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
6052, 59r19.29vva 3207 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) → 𝑧 = (𝐴𝐷𝐵))
61 simpl1 1191 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐷 ∈ (PsMet‘𝑋))
62 simpl2 1192 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴𝑋)
63 psmet0 23661 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)
6461, 62, 63syl2anc 584 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴𝐷𝐴) = 0)
65 relelec 8693 . . . . . . . . . . 11 (Rel → (𝐴 ∈ [𝐴] 𝐴 𝐴))
6661, 34, 653syl 18 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
671a1i 11 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → = (~Met𝐷))
6867breqd 5116 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 𝐴𝐴(~Met𝐷)𝐴))
69 metidv 32473 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐴𝑋)) → (𝐴(~Met𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7061, 62, 62, 69syl12anc 835 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴(~Met𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7166, 68, 703bitrd 304 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∈ [𝐴] ↔ (𝐴𝐷𝐴) = 0))
7264, 71mpbird 256 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴 ∈ [𝐴] )
73 simpl3 1193 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵𝑋)
74 psmet0 23661 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵𝑋) → (𝐵𝐷𝐵) = 0)
7561, 73, 74syl2anc 584 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵𝐷𝐵) = 0)
76 relelec 8693 . . . . . . . . . . 11 (Rel → (𝐵 ∈ [𝐵] 𝐵 𝐵))
7761, 34, 763syl 18 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] 𝐵 𝐵))
7867breqd 5116 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 𝐵𝐵(~Met𝐷)𝐵))
79 metidv 32473 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐵𝑋)) → (𝐵(~Met𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
8061, 73, 73, 79syl12anc 835 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵(~Met𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
8177, 78, 803bitrd 304 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] ↔ (𝐵𝐷𝐵) = 0))
8275, 81mpbird 256 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵 ∈ [𝐵] )
83 simpr 485 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝑧 = (𝐴𝐷𝐵))
84 rspceov 7404 . . . . . . . 8 ((𝐴 ∈ [𝐴] 𝐵 ∈ [𝐵] 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏))
8572, 82, 83, 84syl3anc 1371 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏))
8660, 85impbida 799 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝐴𝐷𝐵)))
8786abbidv 2805 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {𝑧𝑧 = (𝐴𝐷𝐵)})
88 df-sn 4587 . . . . 5 {(𝐴𝐷𝐵)} = {𝑧𝑧 = (𝐴𝐷𝐵)}
8987, 88eqtr4di 2794 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
9089unieqd 4879 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
91 ovex 7390 . . . 4 (𝐴𝐷𝐵) ∈ V
9291unisn 4887 . . 3 {(𝐴𝐷𝐵)} = (𝐴𝐷𝐵)
9390, 92eqtrdi 2792 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = (𝐴𝐷𝐵))
944, 25, 933eqtrd 2780 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  Vcvv 3445  wss 3910  {csn 4586   cuni 4865   class class class wbr 5105   × cxp 5631  Rel wrel 5638  cfv 6496  (class class class)co 7357  cmpo 7359  [cec 8646   / cqs 8647  0cc0 11051  PsMetcpsmet 20780  ~Metcmetid 32467  pstoMetcpstm 32468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-ec 8650  df-qs 8654  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-xadd 13034  df-psmet 20788  df-metid 32469  df-pstm 32470
This theorem is referenced by:  pstmxmet  32478
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