Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-imdirval3 Structured version   Visualization version   GIF version

Theorem bj-imdirval3 36065
Description: Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
Hypotheses
Ref Expression
bj-imdirval3.exa (𝜑𝐴𝑈)
bj-imdirval3.exb (𝜑𝐵𝑉)
bj-imdirval3.arg (𝜑𝑅 ⊆ (𝐴 × 𝐵))
Assertion
Ref Expression
bj-imdirval3 (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))

Proof of Theorem bj-imdirval3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-imdirval3.exa . . . . . 6 (𝜑𝐴𝑈)
2 bj-imdirval3.exb . . . . . 6 (𝜑𝐵𝑉)
3 bj-imdirval3.arg . . . . . 6 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
41, 2, 3bj-imdirval2 36064 . . . . 5 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
54breqd 5160 . . . 4 (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)}𝑌))
6 brabv 5570 . . . 4 (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
75, 6syl6bi 253 . . 3 (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
87pm4.71rd 564 . 2 (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌)))
9 simpl 484 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
109adantl 483 . . . 4 ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → 𝑋 ∈ V)
11 simpr 486 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
1211adantl 483 . . . 4 ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → 𝑌 ∈ V)
134adantr 482 . . . 4 ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
14 simpl 484 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
1514sseq1d 4014 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐴𝑋𝐴))
16 simpr 486 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
1716sseq1d 4014 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑦𝐵𝑌𝐵))
1815, 17anbi12d 632 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥𝐴𝑦𝐵) ↔ (𝑋𝐴𝑌𝐵)))
19 imaeq2 6056 . . . . . . 7 (𝑥 = 𝑋 → (𝑅𝑥) = (𝑅𝑋))
20 id 22 . . . . . . 7 (𝑦 = 𝑌𝑦 = 𝑌)
2119, 20eqeqan12d 2747 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑅𝑥) = 𝑦 ↔ (𝑅𝑋) = 𝑌))
2218, 21anbi12d 632 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
2322adantl 483 . . . 4 (((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
2410, 12, 13, 23brabd 36029 . . 3 ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
2524pm5.32da 580 . 2 (𝜑 → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌))))
26 simpr 486 . . 3 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌))
271adantr 482 . . . . . . . 8 ((𝜑𝑋𝐴) → 𝐴𝑈)
28 simpr 486 . . . . . . . 8 ((𝜑𝑋𝐴) → 𝑋𝐴)
2927, 28ssexd 5325 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋 ∈ V)
3029ex 414 . . . . . 6 (𝜑 → (𝑋𝐴𝑋 ∈ V))
312adantr 482 . . . . . . . 8 ((𝜑𝑌𝐵) → 𝐵𝑉)
32 simpr 486 . . . . . . . 8 ((𝜑𝑌𝐵) → 𝑌𝐵)
3331, 32ssexd 5325 . . . . . . 7 ((𝜑𝑌𝐵) → 𝑌 ∈ V)
3433ex 414 . . . . . 6 (𝜑 → (𝑌𝐵𝑌 ∈ V))
3530, 34anim12d 610 . . . . 5 (𝜑 → ((𝑋𝐴𝑌𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
3635adantrd 493 . . . 4 (𝜑 → (((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
3736ancrd 553 . . 3 (𝜑 → (((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌))))
3826, 37impbid2 225 . 2 (𝜑 → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)) ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
398, 25, 383bitrd 305 1 (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  wss 3949   class class class wbr 5149  {copab 5211   × cxp 5675  cima 5680  cfv 6544  (class class class)co 7409  𝒫*cimdir 36059
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-imdir 36060
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator