Theorem bj-imdirvallem 35347

Description: Lemma for bj-imdirval 35348 and bj-iminvval 35360. (Contributed by BJ, 23-May-2024.)

Hypotheses

Ref	Expression
bj-imdirvallem.1	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-imdirvallem.2	⊢ (𝜑 → 𝐵 ∈ 𝑉)
bj-imdirvallem.df	⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)}))

Assertion

Ref	Expression
bj-imdirvallem	⊢ (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑟,𝑥,𝑦   𝐵,𝑎,𝑏,𝑟,𝑥,𝑦   𝜑,𝑎,𝑏,𝑟   𝜓,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑟)   𝐶(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑈(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)

Proof of Theorem bj-imdirvallem

Step	Hyp	Ref	Expression
1		bj-imdirvallem.df	. . 3 ⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)}))
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)})))
3		xpeq12 5615	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
4	3	pweqd 4558	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
5	4	adantl 482	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
6		sseq2 3952	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑥 ⊆ 𝑎 ↔ 𝑥 ⊆ 𝐴))
7		sseq2 3952	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑦 ⊆ 𝑏 ↔ 𝑦 ⊆ 𝐵))
8	6, 7	bi2anan9 636	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)))
9	8	anbi1d 630	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)))
10	9	opabbidv 5145	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)})
11	10	adantl 482	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)})
12	5, 11	mpteq12dv 5170	. 2 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)}) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}))
13		bj-imdirvallem.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
14	13	elexd 3451	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
15		bj-imdirvallem.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
16	15	elexd 3451	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
17	13, 15	xpexd 7595	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
18	17	pwexd 5306	. . 3 ⊢ (𝜑 → 𝒫 (𝐴 × 𝐵) ∈ V)
19	18	mptexd 7097	. 2 ⊢ (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}) ∈ V)
20	2, 12, 14, 16, 19	ovmpod 7419	1 ⊢ (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  Vcvv 3431   ⊆ wss 3892  𝒫 cpw 4539  {copab 5141   ↦ cmpt 5162   × cxp 5588  (class class class)co 7271   ∈ cmpo 7273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276

This theorem is referenced by:  bj-imdirval  35348  bj-iminvval  35360