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Theorem bj-imdirval3 36368
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 36367 . . . . 5 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
54breqd 5158 . . . 4 (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)}𝑌))
6 brabv 5568 . . . 4 (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
75, 6syl6bi 252 . . 3 (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
87pm4.71rd 561 . 2 (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌)))
9 simpl 481 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
109adantl 480 . . . 4 ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → 𝑋 ∈ V)
11 simpr 483 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
1211adantl 480 . . . 4 ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → 𝑌 ∈ V)
134adantr 479 . . . 4 ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
14 simpl 481 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
1514sseq1d 4012 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐴𝑋𝐴))
16 simpr 483 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
1716sseq1d 4012 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑦𝐵𝑌𝐵))
1815, 17anbi12d 629 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥𝐴𝑦𝐵) ↔ (𝑋𝐴𝑌𝐵)))
19 imaeq2 6054 . . . . . . 7 (𝑥 = 𝑋 → (𝑅𝑥) = (𝑅𝑋))
20 id 22 . . . . . . 7 (𝑦 = 𝑌𝑦 = 𝑌)
2119, 20eqeqan12d 2744 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑅𝑥) = 𝑦 ↔ (𝑅𝑋) = 𝑌))
2218, 21anbi12d 629 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
2322adantl 480 . . . 4 (((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
2410, 12, 13, 23brabd 36332 . . 3 ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
2524pm5.32da 577 . 2 (𝜑 → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌))))
26 simpr 483 . . 3 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌))
271adantr 479 . . . . . . . 8 ((𝜑𝑋𝐴) → 𝐴𝑈)
28 simpr 483 . . . . . . . 8 ((𝜑𝑋𝐴) → 𝑋𝐴)
2927, 28ssexd 5323 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋 ∈ V)
3029ex 411 . . . . . 6 (𝜑 → (𝑋𝐴𝑋 ∈ V))
312adantr 479 . . . . . . . 8 ((𝜑𝑌𝐵) → 𝐵𝑉)
32 simpr 483 . . . . . . . 8 ((𝜑𝑌𝐵) → 𝑌𝐵)
3331, 32ssexd 5323 . . . . . . 7 ((𝜑𝑌𝐵) → 𝑌 ∈ V)
3433ex 411 . . . . . 6 (𝜑 → (𝑌𝐵𝑌 ∈ V))
3530, 34anim12d 607 . . . . 5 (𝜑 → ((𝑋𝐴𝑌𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
3635adantrd 490 . . . 4 (𝜑 → (((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
3736ancrd 550 . . 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 304 1 (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  Vcvv 3472  wss 3947   class class class wbr 5147  {copab 5209   × cxp 5673  cima 5678  cfv 6542  (class class class)co 7411  𝒫*cimdir 36362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-imdir 36363
This theorem is referenced by: (None)
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