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Theorem pstmfval 32541
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 32540 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (pstoMetβ€˜π·) = (π‘₯ ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)}))
323ad2ant1 1134 . . 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 7378 . 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 6860 . . . . 5 ∼ ∈ V
65ecelqsi 8718 . . . 4 (𝐴 ∈ 𝑋 β†’ [𝐴] ∼ ∈ (𝑋 / ∼ ))
763ad2ant2 1135 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ [𝐴] ∼ ∈ (𝑋 / ∼ ))
85ecelqsi 8718 . . . 4 (𝐡 ∈ 𝑋 β†’ [𝐡] ∼ ∈ (𝑋 / ∼ ))
983ad2ant3 1136 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ [𝐡] ∼ ∈ (𝑋 / ∼ ))
10 rexeq 3309 . . . . . 6 (π‘₯ = [𝐴] ∼ β†’ (βˆƒπ‘Ž ∈ π‘₯ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘) ↔ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘)))
1110abbidv 2802 . . . . 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 4883 . . . 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 3309 . . . . . . 7 (𝑦 = [𝐡] ∼ β†’ (βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘) ↔ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)))
1413rexbidv 3172 . . . . . 6 (𝑦 = [𝐡] ∼ β†’ (βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ 𝑦 𝑧 = (π‘Žπ·π‘) ↔ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)))
1514abbidv 2802 . . . . 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 4883 . . . 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 2733 . . . 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 8658 . . . . . . 7 ( ∼ ∈ V β†’ [𝐴] ∼ ∈ V)
195, 18ax-mp 5 . . . . . 6 [𝐴] ∼ ∈ V
20 ecexg 8658 . . . . . . 7 ( ∼ ∈ V β†’ [𝐡] ∼ ∈ V)
215, 20ax-mp 5 . . . . . 6 [𝐡] ∼ ∈ V
2219, 21ab2rexex 7916 . . . . 5 {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} ∈ V
2322uniex 7682 . . . 4 βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} ∈ V
2412, 16, 17, 23ovmpo 7519 . . 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 585 . 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 1197 . . . . . . . . . . 11 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝑧 = (𝑒𝐷𝑓))
27 simpl1 1192 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
28 simpr1 1195 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝑒 ∈ [𝐴] ∼ )
29 metidss 32536 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋))
301, 29eqsstrid 3996 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ∼ βŠ† (𝑋 Γ— 𝑋))
31 xpss 5653 . . . . . . . . . . . . . . . . . . 19 (𝑋 Γ— 𝑋) βŠ† (V Γ— V)
3230, 31sstrdi 3960 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ∼ βŠ† (V Γ— V))
33 df-rel 5644 . . . . . . . . . . . . . . . . . 18 (Rel ∼ ↔ ∼ βŠ† (V Γ— V))
3432, 33sylibr 233 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ Rel ∼ )
35343ad2ant1 1134 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ Rel ∼ )
3635adantr 482 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ Rel ∼ )
37 relelec 8699 . . . . . . . . . . . . . . 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 5115 . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . 14 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝑓 ∈ [𝐡] ∼ )
43 relelec 8699 . . . . . . . . . . . . . . 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 5115 . . . . . . . . . . . . 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 32538 . . . . . . . . . . . 12 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴(~Metβ€˜π·)𝑒 ∧ 𝐡(~Metβ€˜π·)𝑓)) β†’ (𝐴𝐷𝐡) = (𝑒𝐷𝑓))
4927, 41, 47, 48syl12anc 836 . . . . . . . . . . 11 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ (𝐴𝐷𝐡) = (𝑒𝐷𝑓))
5026, 49eqtr4d 2776 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐡] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) β†’ 𝑧 = (𝐴𝐷𝐡))
5150adantlr 714 . . . . . . . . 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 1361 . . . . . . . 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 7368 . . . . . . . . . . . 12 (π‘Ž = 𝑒 β†’ (π‘Žπ·π‘) = (𝑒𝐷𝑏))
5453eqeq2d 2744 . . . . . . . . . . 11 (π‘Ž = 𝑒 β†’ (𝑧 = (π‘Žπ·π‘) ↔ 𝑧 = (𝑒𝐷𝑏)))
55 oveq2 7369 . . . . . . . . . . . 12 (𝑏 = 𝑓 β†’ (𝑒𝐷𝑏) = (𝑒𝐷𝑓))
5655eqeq2d 2744 . . . . . . . . . . 11 (𝑏 = 𝑓 β†’ (𝑧 = (𝑒𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑓)))
5754, 56cbvrex2vw 3227 . . . . . . . . . 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 483 . . . . . . . 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 3204 . . . . . . 7 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)) β†’ 𝑧 = (𝐴𝐷𝐡))
61 simpl1 1192 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
62 simpl2 1193 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝐴 ∈ 𝑋)
63 psmet0 23684 . . . . . . . . . 10 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐷𝐴) = 0)
6461, 62, 63syl2anc 585 . . . . . . . . 9 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐴𝐷𝐴) = 0)
65 relelec 8699 . . . . . . . . . . 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 5120 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐴 ∼ 𝐴 ↔ 𝐴(~Metβ€˜π·)𝐴))
69 metidv 32537 . . . . . . . . . . 11 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) β†’ (𝐴(~Metβ€˜π·)𝐴 ↔ (𝐴𝐷𝐴) = 0))
7061, 62, 62, 69syl12anc 836 . . . . . . . . . 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 1194 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝐡 ∈ 𝑋)
74 psmet0 23684 . . . . . . . . . 10 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐡 ∈ 𝑋) β†’ (𝐡𝐷𝐡) = 0)
7561, 73, 74syl2anc 585 . . . . . . . . 9 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐡𝐷𝐡) = 0)
76 relelec 8699 . . . . . . . . . . 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 5120 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ (𝐡 ∼ 𝐡 ↔ 𝐡(~Metβ€˜π·)𝐡))
79 metidv 32537 . . . . . . . . . . 11 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐡 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐡(~Metβ€˜π·)𝐡 ↔ (𝐡𝐷𝐡) = 0))
8061, 73, 73, 79syl12anc 836 . . . . . . . . . 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 486 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ 𝑧 = (𝐴𝐷𝐡))
84 rspceov 7408 . . . . . . . 8 ((𝐴 ∈ [𝐴] ∼ ∧ 𝐡 ∈ [𝐡] ∼ ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘))
8572, 82, 83, 84syl3anc 1372 . . . . . . 7 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐡)) β†’ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘))
8660, 85impbida 800 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘) ↔ 𝑧 = (𝐴𝐷𝐡)))
8786abbidv 2802 . . . . 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 4591 . . . . 5 {(𝐴𝐷𝐡)} = {𝑧 ∣ 𝑧 = (𝐴𝐷𝐡)}
8987, 88eqtr4di 2791 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} = {(𝐴𝐷𝐡)})
9089unieqd 4883 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} = βˆͺ {(𝐴𝐷𝐡)})
91 ovex 7394 . . . 4 (𝐴𝐷𝐡) ∈ V
9291unisn 4891 . . 3 βˆͺ {(𝐴𝐷𝐡)} = (𝐴𝐷𝐡)
9390, 92eqtrdi 2789 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ βˆͺ {𝑧 ∣ βˆƒπ‘Ž ∈ [ 𝐴] ∼ βˆƒπ‘ ∈ [ 𝐡] ∼ 𝑧 = (π‘Žπ·π‘)} = (𝐴𝐷𝐡))
944, 25, 933eqtrd 2777 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3070  Vcvv 3447   βŠ† wss 3914  {csn 4590  βˆͺ cuni 4869   class class class wbr 5109   Γ— cxp 5635  Rel wrel 5642  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  [cec 8652   / cqs 8653  0cc0 11059  PsMetcpsmet 20803  ~Metcmetid 32531  pstoMetcpstm 32532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-er 8654  df-ec 8656  df-qs 8660  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-xadd 13042  df-psmet 20811  df-metid 32533  df-pstm 32534
This theorem is referenced by:  pstmxmet  32542
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