Theorem bj-iminvval2 37858

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

Hypotheses

Ref	Expression
bj-iminvval2.exa	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-iminvval2.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
bj-iminvval2.arg	⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))

Assertion

Ref	Expression
bj-iminvval2	⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 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-iminvval2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-iminvval2.exa	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		bj-iminvval2.exb	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3	1, 2	bj-iminvval 37857	. 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))
4		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
5	4	cnveqd 5861	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ◡𝑟 = ◡𝑅)
6	5	imaeq1d 6061	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (◡𝑟 “ 𝑦) = (◡𝑅 “ 𝑦))
7	6	eqeq2d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑥 = (◡𝑟 “ 𝑦) ↔ 𝑥 = (◡𝑅 “ 𝑦)))
8	7	anbi2d 641	. . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦)) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))))
9	8	opabbidv 5177	. 2 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))})
10	1, 2	xpexd 7746	. . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
11		bj-iminvval2.arg	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
12	10, 11	sselpwd 5299	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
13	1, 2	bj-imdirval2lem 37846	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))} ∈ V)
14	3, 9, 12, 13	fvmptd 6997	1 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562  {copab 5173   × cxp 5659  ^◡ccnv 5660   “ cima 5664  ‘cfv 6536  (class class class)co 7410  𝒫^*ciminv 37855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-iminv 37856

This theorem is referenced by:  bj-iminvid  37859