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Theorem pstmfval 31420
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 31419 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
323ad2ant1 1134 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
43oveqd 7189 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ))
51fvexi 6690 . . . . 5 ∈ V
65ecelqsi 8386 . . . 4 (𝐴𝑋 → [𝐴] ∈ (𝑋 / ))
763ad2ant2 1135 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → [𝐴] ∈ (𝑋 / ))
85ecelqsi 8386 . . . 4 (𝐵𝑋 → [𝐵] ∈ (𝑋 / ))
983ad2ant3 1136 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → [𝐵] ∈ (𝑋 / ))
10 rexeq 3311 . . . . . 6 (𝑥 = [𝐴] → (∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)))
1110abbidv 2802 . . . . 5 (𝑥 = [𝐴] → {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
1211unieqd 4810 . . . 4 (𝑥 = [𝐴] {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
13 rexeq 3311 . . . . . . 7 (𝑦 = [𝐵] → (∃𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)))
1413rexbidv 3207 . . . . . 6 (𝑦 = [𝐵] → (∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)))
1514abbidv 2802 . . . . 5 (𝑦 = [𝐵] → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
1615unieqd 4810 . . . 4 (𝑦 = [𝐵] {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
17 eqid 2738 . . . 4 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
18 ecexg 8326 . . . . . . 7 ( ∈ V → [𝐴] ∈ V)
195, 18ax-mp 5 . . . . . 6 [𝐴] ∈ V
20 ecexg 8326 . . . . . . 7 ( ∈ V → [𝐵] ∈ V)
215, 20ax-mp 5 . . . . . 6 [𝐵] ∈ V
2219, 21ab2rexex 7707 . . . . 5 {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} ∈ V
2322uniex 7487 . . . 4 {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} ∈ V
2412, 16, 17, 23ovmpo 7327 . . 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 587 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ) = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)})
26 simpr3 1197 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝑒𝐷𝑓))
27 simpl1 1192 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐷 ∈ (PsMet‘𝑋))
28 simpr1 1195 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑒 ∈ [𝐴] )
29 metidss 31415 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
301, 29eqsstrid 3925 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ (PsMet‘𝑋) → ⊆ (𝑋 × 𝑋))
31 xpss 5541 . . . . . . . . . . . . . . . . . . 19 (𝑋 × 𝑋) ⊆ (V × V)
3230, 31sstrdi 3889 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ (PsMet‘𝑋) → ⊆ (V × V))
33 df-rel 5532 . . . . . . . . . . . . . . . . . 18 (Rel ⊆ (V × V))
3432, 33sylibr 237 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (PsMet‘𝑋) → Rel )
35343ad2ant1 1134 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → Rel )
3635adantr 484 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → Rel )
37 relelec 8367 . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐴 𝑒)
401breqi 5036 . . . . . . . . . . . . 13 (𝐴 𝑒𝐴(~Met𝐷)𝑒)
4139, 40sylib 221 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐴(~Met𝐷)𝑒)
42 simpr2 1196 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑓 ∈ [𝐵] )
43 relelec 8367 . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐵 𝑓)
461breqi 5036 . . . . . . . . . . . . 13 (𝐵 𝑓𝐵(~Met𝐷)𝑓)
4745, 46sylib 221 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝐵(~Met𝐷)𝑓)
48 metideq 31417 . . . . . . . . . . . 12 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝑒𝐵(~Met𝐷)𝑓)) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
4927, 41, 47, 48syl12anc 836 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
5026, 49eqtr4d 2776 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
5150adantlr 715 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) ∧ (𝑒 ∈ [𝐴] 𝑓 ∈ [𝐵] 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
52513anassrs 1361 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) ∧ 𝑒 ∈ [𝐴] ) ∧ 𝑓 ∈ [𝐵] ) ∧ 𝑧 = (𝑒𝐷𝑓)) → 𝑧 = (𝐴𝐷𝐵))
53 oveq1 7179 . . . . . . . . . . . 12 (𝑎 = 𝑒 → (𝑎𝐷𝑏) = (𝑒𝐷𝑏))
5453eqeq2d 2749 . . . . . . . . . . 11 (𝑎 = 𝑒 → (𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑏)))
55 oveq2 7180 . . . . . . . . . . . 12 (𝑏 = 𝑓 → (𝑒𝐷𝑏) = (𝑒𝐷𝑓))
5655eqeq2d 2749 . . . . . . . . . . 11 (𝑏 = 𝑓 → (𝑧 = (𝑒𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑓)))
5754, 56cbvrex2vw 3363 . . . . . . . . . 10 (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
5857biimpi 219 . . . . . . . . 9 (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) → ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
5958adantl 485 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) → ∃𝑒 ∈ [ 𝐴] 𝑓 ∈ [ 𝐵] 𝑧 = (𝑒𝐷𝑓))
6052, 59r19.29vva 3243 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)) → 𝑧 = (𝐴𝐷𝐵))
61 simpl1 1192 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐷 ∈ (PsMet‘𝑋))
62 simpl2 1193 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴𝑋)
63 psmet0 23063 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)
6461, 62, 63syl2anc 587 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴𝐷𝐴) = 0)
65 relelec 8367 . . . . . . . . . . 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 5041 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 𝐴𝐴(~Met𝐷)𝐴))
69 metidv 31416 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐴𝑋)) → (𝐴(~Met𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7061, 62, 62, 69syl12anc 836 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴(~Met𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7166, 68, 703bitrd 308 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∈ [𝐴] ↔ (𝐴𝐷𝐴) = 0))
7264, 71mpbird 260 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴 ∈ [𝐴] )
73 simpl3 1194 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵𝑋)
74 psmet0 23063 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵𝑋) → (𝐵𝐷𝐵) = 0)
7561, 73, 74syl2anc 587 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵𝐷𝐵) = 0)
76 relelec 8367 . . . . . . . . . . 11 (Rel → (𝐵 ∈ [𝐵] 𝐵 𝐵))
7761, 34, 763syl 18 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] 𝐵 𝐵))
7867breqd 5041 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 𝐵𝐵(~Met𝐷)𝐵))
79 metidv 31416 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵𝑋𝐵𝑋)) → (𝐵(~Met𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
8061, 73, 73, 79syl12anc 836 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵(~Met𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
8177, 78, 803bitrd 308 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] ↔ (𝐵𝐷𝐵) = 0))
8275, 81mpbird 260 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵 ∈ [𝐵] )
83 simpr 488 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝑧 = (𝐴𝐷𝐵))
84 rspceov 7219 . . . . . . . 8 ((𝐴 ∈ [𝐴] 𝐵 ∈ [𝐵] 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏))
8572, 82, 83, 84syl3anc 1372 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏))
8660, 85impbida 801 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝐴𝐷𝐵)))
8786abbidv 2802 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {𝑧𝑧 = (𝐴𝐷𝐵)})
88 df-sn 4517 . . . . 5 {(𝐴𝐷𝐵)} = {𝑧𝑧 = (𝐴𝐷𝐵)}
8987, 88eqtr4di 2791 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
9089unieqd 4810 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
91 ovex 7205 . . . 4 (𝐴𝐷𝐵) ∈ V
9291unisn 4818 . . 3 {(𝐴𝐷𝐵)} = (𝐴𝐷𝐵)
9390, 92eqtrdi 2789 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] 𝑏 ∈ [ 𝐵] 𝑧 = (𝑎𝐷𝑏)} = (𝐴𝐷𝐵))
944, 25, 933eqtrd 2777 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  {cab 2716  wrex 3054  Vcvv 3398  wss 3843  {csn 4516   cuni 4796   class class class wbr 5030   × cxp 5523  Rel wrel 5530  cfv 6339  (class class class)co 7172  cmpo 7174  [cec 8320   / cqs 8321  0cc0 10617  PsMetcpsmet 20203  ~Metcmetid 31410  pstoMetcpstm 31411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-cnex 10673  ax-resscn 10674  ax-1cn 10675  ax-icn 10676  ax-addcl 10677  ax-addrcl 10678  ax-mulcl 10679  ax-mulrcl 10680  ax-mulcom 10681  ax-addass 10682  ax-mulass 10683  ax-distr 10684  ax-i2m1 10685  ax-1ne0 10686  ax-1rid 10687  ax-rnegex 10688  ax-rrecex 10689  ax-cnre 10690  ax-pre-lttri 10691  ax-pre-lttrn 10692  ax-pre-ltadd 10693
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-po 5442  df-so 5443  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7175  df-oprab 7176  df-mpo 7177  df-1st 7716  df-2nd 7717  df-er 8322  df-ec 8324  df-qs 8328  df-map 8441  df-en 8558  df-dom 8559  df-sdom 8560  df-pnf 10757  df-mnf 10758  df-xr 10759  df-ltxr 10760  df-le 10761  df-xadd 12593  df-psmet 20211  df-metid 31412  df-pstm 31413
This theorem is referenced by:  pstmxmet  31421
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