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Theorem rfovcnvfvd 43247
Description: Value of the converse of the operator, (𝐴𝑂𝐡), which maps between relations and functions for relations between base sets, 𝐴 and 𝐡, evaluated at function 𝐺. (Contributed by RP, 27-Apr-2021.)
Hypotheses
Ref Expression
rfovd.rf 𝑂 = (π‘Ž ∈ V, 𝑏 ∈ V ↦ (π‘Ÿ ∈ 𝒫 (π‘Ž Γ— 𝑏) ↦ (π‘₯ ∈ π‘Ž ↦ {𝑦 ∈ 𝑏 ∣ π‘₯π‘Ÿπ‘¦})))
rfovd.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
rfovd.b (πœ‘ β†’ 𝐡 ∈ π‘Š)
rfovcnvf1od.f 𝐹 = (𝐴𝑂𝐡)
rfovcnvfv.g (πœ‘ β†’ 𝐺 ∈ (𝒫 𝐡 ↑m 𝐴))
Assertion
Ref Expression
rfovcnvfvd (πœ‘ β†’ (β—‘πΉβ€˜πΊ) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (πΊβ€˜π‘₯))})
Distinct variable groups:   𝐴,π‘Ž,𝑏,π‘Ÿ,π‘₯,𝑦   𝐡,π‘Ž,𝑏,π‘Ÿ,π‘₯,𝑦   π‘₯,𝐺,𝑦   π‘Š,π‘Ž,π‘₯   πœ‘,π‘Ž,𝑏,π‘Ÿ,π‘₯,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦,π‘Ÿ,π‘Ž,𝑏)   𝐺(π‘Ÿ,π‘Ž,𝑏)   𝑂(π‘₯,𝑦,π‘Ÿ,π‘Ž,𝑏)   𝑉(π‘₯,𝑦,π‘Ÿ,π‘Ž,𝑏)   π‘Š(𝑦,π‘Ÿ,𝑏)

Proof of Theorem rfovcnvfvd
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 rfovd.rf . . 3 𝑂 = (π‘Ž ∈ V, 𝑏 ∈ V ↦ (π‘Ÿ ∈ 𝒫 (π‘Ž Γ— 𝑏) ↦ (π‘₯ ∈ π‘Ž ↦ {𝑦 ∈ 𝑏 ∣ π‘₯π‘Ÿπ‘¦})))
2 rfovd.a . . 3 (πœ‘ β†’ 𝐴 ∈ 𝑉)
3 rfovd.b . . 3 (πœ‘ β†’ 𝐡 ∈ π‘Š)
4 rfovcnvf1od.f . . 3 𝐹 = (𝐴𝑂𝐡)
51, 2, 3, 4rfovcnvd 43245 . 2 (πœ‘ β†’ ◑𝐹 = (𝑔 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘”β€˜π‘₯))}))
6 fveq1 6880 . . . . . 6 (𝑔 = 𝐺 β†’ (π‘”β€˜π‘₯) = (πΊβ€˜π‘₯))
76eleq2d 2811 . . . . 5 (𝑔 = 𝐺 β†’ (𝑦 ∈ (π‘”β€˜π‘₯) ↔ 𝑦 ∈ (πΊβ€˜π‘₯)))
87anbi2d 628 . . . 4 (𝑔 = 𝐺 β†’ ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘”β€˜π‘₯)) ↔ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (πΊβ€˜π‘₯))))
98opabbidv 5204 . . 3 (𝑔 = 𝐺 β†’ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘”β€˜π‘₯))} = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (πΊβ€˜π‘₯))})
109adantl 481 . 2 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘”β€˜π‘₯))} = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (πΊβ€˜π‘₯))})
11 rfovcnvfv.g . 2 (πœ‘ β†’ 𝐺 ∈ (𝒫 𝐡 ↑m 𝐴))
12 simprl 768 . . 3 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (πΊβ€˜π‘₯))) β†’ π‘₯ ∈ 𝐴)
13 elmapi 8839 . . . . . . . 8 (𝐺 ∈ (𝒫 𝐡 ↑m 𝐴) β†’ 𝐺:π΄βŸΆπ’« 𝐡)
1413ffvelcdmda 7076 . . . . . . 7 ((𝐺 ∈ (𝒫 𝐡 ↑m 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ (πΊβ€˜π‘₯) ∈ 𝒫 𝐡)
1511, 14sylan 579 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (πΊβ€˜π‘₯) ∈ 𝒫 𝐡)
1615elpwid 4603 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (πΊβ€˜π‘₯) βŠ† 𝐡)
1716sseld 3973 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (𝑦 ∈ (πΊβ€˜π‘₯) β†’ 𝑦 ∈ 𝐡))
1817impr 454 . . 3 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (πΊβ€˜π‘₯))) β†’ 𝑦 ∈ 𝐡)
192, 3, 12, 18opabex2 8036 . 2 (πœ‘ β†’ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (πΊβ€˜π‘₯))} ∈ V)
205, 10, 11, 19fvmptd 6995 1 (πœ‘ β†’ (β—‘πΉβ€˜πΊ) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (πΊβ€˜π‘₯))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3424  Vcvv 3466  π’« cpw 4594   class class class wbr 5138  {copab 5200   ↦ cmpt 5221   Γ— cxp 5664  β—‘ccnv 5665  β€˜cfv 6533  (class class class)co 7401   ∈ cmpo 7403   ↑m cmap 8816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8818
This theorem is referenced by: (None)
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