Theorem bj-imdirval2 36585

Description: Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)

Hypotheses

Ref	Expression
bj-imdirval2.exa	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-imdirval2.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
bj-imdirval2.arg	⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))

Assertion

Ref	Expression
bj-imdirval2	⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem bj-imdirval2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-imdirval2.exa	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		bj-imdirval2.exb	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3	1, 2	bj-imdirval 36583	. 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)}))
4		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
5	4	imaeq1d 6056	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑟 “ 𝑥) = (𝑅 “ 𝑥))
6	5	eqeq1d 2729	. . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ((𝑟 “ 𝑥) = 𝑦 ↔ (𝑅 “ 𝑥) = 𝑦))
7	6	anbi2d 628	. . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)))
8	7	opabbidv 5208	. 2 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
9	1, 2	xpexd 7745	. . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
10		bj-imdirval2.arg	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
11	9, 10	sselpwd 5322	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
12	1, 2	bj-imdirval2lem 36584	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)} ∈ V)
13	3, 8, 11, 12	fvmptd 7006	1 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ⊆ wss 3944  𝒫 cpw 4598  {copab 5204   × cxp 5670   “ cima 5675  ‘cfv 6542  (class class class)co 7414  𝒫_*cimdir 36580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7417  df-oprab 7418  df-mpo 7419  df-imdir 36581

This theorem is referenced by:  bj-imdirval3  36586  bj-imdirid  36588  bj-imdirco  36592