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Theorem pstmfval 33175
Description: Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
pstmval.1 ∼ = (~Metβ€˜π·)
Assertion
Ref Expression
pstmfval ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ([𝐴] ∼ (pstoMetβ€˜π·)[𝐡] ∼ ) = (𝐴𝐷𝐡))

Proof of Theorem pstmfval
Dummy variables π‘Ž 𝑏 π‘₯ 𝑦 𝑧 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pstmval.1 . . . . 5 ∼ = (~Metβ€˜π·)
21pstmval 33174 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (pstoMetβ€˜π·) = (π‘₯ ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)}))
323ad2ant1 1132 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (pstoMetβ€˜π·) = (π‘₯ ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)}))
43oveqd 7429 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ([𝐴] ∼ (pstoMetβ€˜π·)[𝐡] ∼ ) = ([𝐴] ∼ (π‘₯ ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)})[𝐡] ∼ ))
51fvexi 6905 . . . . 5 ∼ ∈ V
65ecelqsi 8771 . . . 4 (𝐴 ∈ 𝑋 β†’ [𝐴] ∼ ∈ (𝑋 / ∼ ))
763ad2ant2 1133 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ [𝐴] ∼ ∈ (𝑋 / ∼ ))
85ecelqsi 8771 . . . 4 (𝐡 ∈ 𝑋 β†’ [𝐡] ∼ ∈ (𝑋 / ∼ ))
983ad2ant3 1134 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ [𝐡] ∼ ∈ (𝑋 / ∼ ))
10 rexeq 3320 . . . . . 6 (π‘₯ = [𝐴] ∼ β†’ (βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘) ↔ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)))
1110abbidv 2800 . . . . 5 (π‘₯ = [𝐴] ∼ β†’ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)} = {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)})
1211unieqd 4922 . . . 4 (π‘₯ = [𝐴] ∼ β†’ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)} = βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)})
13 rexeq 3320 . . . . . . 7 (𝑦 = [𝐡] ∼ β†’ (βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘) ↔ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)))
1413rexbidv 3177 . . . . . 6 (𝑦 = [𝐡] ∼ β†’ (βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘) ↔ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)))
1514abbidv 2800 . . . . 5 (𝑦 = [𝐡] ∼ β†’ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)} = {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)})
1615unieqd 4922 . . . 4 (𝑦 = [𝐡] ∼ β†’ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)} = βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)})
17 eqid 2731 . . . 4 (π‘₯ ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)}) = (π‘₯ ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)})
18 ecexg 8711 . . . . . . 7 ( ∼ ∈ V β†’ [𝐴] ∼ ∈ V)
195, 18ax-mp 5 . . . . . 6 [𝐴] ∼ ∈ V
20 ecexg 8711 . . . . . . 7 ( ∼ ∈ V β†’ [𝐡] ∼ ∈ V)
215, 20ax-mp 5 . . . . . 6 [𝐡] ∼ ∈ V
2219, 21ab2rexex 7970 . . . . 5 {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} ∈ V
2322uniex 7735 . . . 4 βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} ∈ V
2412, 16, 17, 23ovmpo 7571 . . 3 (([𝐴] ∼ ∈ (𝑋 / ∼ ) ∧ [𝐡] ∼ ∈ (𝑋 / ∼ )) β†’ ([𝐴] ∼ (π‘₯ ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)})[𝐡] ∼ ) = βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)})
257, 9, 24syl2anc 583 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ([𝐴] ∼ (π‘₯ ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)})[𝐡] ∼ ) = βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)})
26 simpr3 1195 . . . . . . . . . . 11 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝑧 = (𝑒𝐷𝑓))
27 simpl1 1190 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
28 simpr1 1193 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝑒 ∈ [𝐴] ∼ )
29 metidss 33170 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋))
301, 29eqsstrid 4030 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ∼ βŠ† (𝑋 Γ— 𝑋))
31 xpss 5692 . . . . . . . . . . . . . . . . . . 19 (𝑋 Γ— 𝑋) βŠ† (V Γ— V)
3230, 31sstrdi 3994 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ∼ βŠ† (V Γ— V))
33 df-rel 5683 . . . . . . . . . . . . . . . . . 18 (Rel ∼ ↔ ∼ βŠ† (V Γ— V))
3432, 33sylibr 233 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ Rel ∼ )
35343ad2ant1 1132 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ Rel ∼ )
3635adantr 480 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ Rel ∼ )
37 relelec 8752 . . . . . . . . . . . . . . 15 (Rel ∼ β†’ (𝑒 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑒))
3836, 37syl 17 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ (𝑒 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑒))
3928, 38mpbid 231 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝐴 ∼ 𝑒)
401breqi 5154 . . . . . . . . . . . . 13 (𝐴 ∼ 𝑒 ↔ 𝐴(~Metβ€˜π·)𝑒)
4139, 40sylib 217 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝐴(~Metβ€˜π·)𝑒)
42 simpr2 1194 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝑓 ∈ [𝐡] ∼ )
43 relelec 8752 . . . . . . . . . . . . . . 15 (Rel ∼ β†’ (𝑓 ∈ [𝐡] ∼ ↔ 𝐡 ∼ 𝑓))
4436, 43syl 17 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ (𝑓 ∈ [𝐡] ∼ ↔ 𝐡 ∼ 𝑓))
4542, 44mpbid 231 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝐡 ∼ 𝑓)
461breqi 5154 . . . . . . . . . . . . 13 (𝐡 ∼ 𝑓 ↔ 𝐡(~Metβ€˜π·)𝑓)
4745, 46sylib 217 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝐡(~Metβ€˜π·)𝑓)
48 metideq 33172 . . . . . . . . . . . 12 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝑒 ∧ 𝐡(~Metβ€˜π·)𝑓)) β†’ (𝐴𝐷𝐡) = (𝑒𝐷𝑓))
4927, 41, 47, 48syl12anc 834 . . . . . . . . . . 11 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ (𝐴𝐷𝐡) = (𝑒𝐷𝑓))
5026, 49eqtr4d 2774 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝑧 = (𝐴𝐷𝐡))
5150adantlr 712 . . . . . . . . 9 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝑧 = (𝐴𝐷𝐡))
52513anassrs 1359 . . . . . . . 8 ((((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)) ∧ 𝑒 ∈ [𝐴] ∼ ) ∧ 𝑓 ∈ [𝐡] ∼ ) ∧ 𝑧 = (𝑒𝐷𝑓)) β†’ 𝑧 = (𝐴𝐷𝐡))
53 oveq1 7419 . . . . . . . . . . . 12 (π‘Ž = 𝑒 β†’ (π‘Žπ·π‘) = (𝑒𝐷𝑏))
5453eqeq2d 2742 . . . . . . . . . . 11 (π‘Ž = 𝑒 β†’ (𝑧 = (π‘Žπ·π‘) ↔ 𝑧 = (𝑒𝐷𝑏)))
55 oveq2 7420 . . . . . . . . . . . 12 (𝑏 = 𝑓 β†’ (𝑒𝐷𝑏) = (𝑒𝐷𝑓))
5655eqeq2d 2742 . . . . . . . . . . 11 (𝑏 = 𝑓 β†’ (𝑧 = (𝑒𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑓)))
5754, 56cbvrex2vw 3238 . . . . . . . . . 10 (βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘) ↔ βˆƒπ‘’ ∈ [ 𝐴] ∼ βˆƒπ‘“ ∈ [ 𝐡] ∼ 𝑧 = (𝑒𝐷𝑓))
5857biimpi 215 . . . . . . . . 9 (βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘) β†’ βˆƒπ‘’ ∈ [ 𝐴] ∼ βˆƒπ‘“ ∈ [ 𝐡] ∼ 𝑧 = (𝑒𝐷𝑓))
5958adantl 481 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)) β†’ βˆƒπ‘’ ∈ [ 𝐴] ∼ βˆƒπ‘“ ∈ [ 𝐡] ∼ 𝑧 = (𝑒𝐷𝑓))
6052, 59r19.29vva 3212 . . . . . . 7 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)) β†’ 𝑧 = (𝐴𝐷𝐡))
61 simpl1 1190 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
62 simpl2 1191 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝐴 ∈ 𝑋)
63 psmet0 24035 . . . . . . . . . 10 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐷𝐴) = 0)
6461, 62, 63syl2anc 583 . . . . . . . . 9 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐴𝐷𝐴) = 0)
65 relelec 8752 . . . . . . . . . . 11 (Rel ∼ β†’ (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
6661, 34, 653syl 18 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
671a1i 11 . . . . . . . . . . 11 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ ∼ = (~Metβ€˜π·))
6867breqd 5159 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐴 ∼ 𝐴 ↔ 𝐴(~Metβ€˜π·)𝐴))
69 metidv 33171 . . . . . . . . . . 11 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) β†’ (𝐴(~Metβ€˜π·)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7061, 62, 62, 69syl12anc 834 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐴(~Metβ€˜π·)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7166, 68, 703bitrd 305 . . . . . . . . 9 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐴 ∈ [𝐴] ∼ ↔ (𝐴𝐷𝐴) = 0))
7264, 71mpbird 257 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝐴 ∈ [𝐴] ∼ )
73 simpl3 1192 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝐡 ∈ 𝑋)
74 psmet0 24035 . . . . . . . . . 10 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐡 ∈ 𝑋) β†’ (𝐡𝐷𝐡) = 0)
7561, 73, 74syl2anc 583 . . . . . . . . 9 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐡𝐷𝐡) = 0)
76 relelec 8752 . . . . . . . . . . 11 (Rel ∼ β†’ (𝐡 ∈ [𝐡] ∼ ↔ 𝐡 ∼ 𝐡))
7761, 34, 763syl 18 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐡 ∈ [𝐡] ∼ ↔ 𝐡 ∼ 𝐡))
7867breqd 5159 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐡 ∼ 𝐡 ↔ 𝐡(~Metβ€˜π·)𝐡))
79 metidv 33171 . . . . . . . . . . 11 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐡 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐡(~Metβ€˜π·)𝐡 ↔ (𝐡𝐷𝐡) = 0))
8061, 73, 73, 79syl12anc 834 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐡(~Metβ€˜π·)𝐡 ↔ (𝐡𝐷𝐡) = 0))
8177, 78, 803bitrd 305 . . . . . . . . 9 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐡 ∈ [𝐡] ∼ ↔ (𝐡𝐷𝐡) = 0))
8275, 81mpbird 257 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝐡 ∈ [𝐡] ∼ )
83 simpr 484 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝑧 = (𝐴𝐷𝐡))
84 rspceov 7459 . . . . . . . 8 ((𝐴 ∈ [𝐴] ∼ ∧ 𝐡 ∈ [𝐡] ∼ ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘))
8572, 82, 83, 84syl3anc 1370 . . . . . . 7 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘))
8660, 85impbida 798 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘) ↔ 𝑧 = (𝐴𝐷𝐡)))
8786abbidv 2800 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} = {𝑧 ∣ 𝑧 = (𝐴𝐷𝐡)})
88 df-sn 4629 . . . . 5 {(𝐴𝐷𝐡)} = {𝑧 ∣ 𝑧 = (𝐴𝐷𝐡)}
8987, 88eqtr4di 2789 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} = {(𝐴𝐷𝐡)})
9089unieqd 4922 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} = βˆͺ {(𝐴𝐷𝐡)})
91 ovex 7445 . . . 4 (𝐴𝐷𝐡) ∈ V
9291unisn 4930 . . 3 βˆͺ {(𝐴𝐷𝐡)} = (𝐴𝐷𝐡)
9390, 92eqtrdi 2787 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} = (𝐴𝐷𝐡))
944, 25, 933eqtrd 2775 1 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ([𝐴] ∼ (pstoMetβ€˜π·)[𝐡] ∼ ) = (𝐴𝐷𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {cab 2708  βˆƒwrex 3069  Vcvv 3473   βŠ† wss 3948  {csn 4628  βˆͺ cuni 4908   class class class wbr 5148   Γ— cxp 5674  Rel wrel 5681  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  [cec 8705   / cqs 8706  0cc0 11114  PsMetcpsmet 21129  ~Metcmetid 33165  pstoMetcpstm 33166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-er 8707  df-ec 8709  df-qs 8713  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-xadd 13098  df-psmet 21137  df-metid 33167  df-pstm 33168
This theorem is referenced by:  pstmxmet  33176
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