Theorem bj-iminvval 38157

Description: Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.)

Hypotheses

Ref	Expression
bj-iminvval.1	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-iminvval.2	⊢ (𝜑 → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
bj-iminvval	⊢ (𝜑 → (𝐴 𝒫* 𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))

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

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

Proof of Theorem bj-iminvval

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-iminvval.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		bj-iminvval.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3		df-iminv 38156	. 2 ⊢ 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))
4	1, 2, 3	bj-imdirvallem 38144	1 ⊢ (𝜑 → (𝐴 𝒫* 𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895  𝒫 cpw 4556  {copab 5166   ↦ cmpt 5185   × cxp 5648  ^◡ccnv 5649   “ cima 5653  (class class class)co 7420   𝒫^* ciminv 38155

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-iminv 38156

This theorem is used by:  bj-iminvval2  38158