Theorem bj-imdirvallem 37385

Description: Lemma for bj-imdirval 37386 and bj-iminvval 37398. (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 5649	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
4	3	pweqd 4571	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
5	4	adantl 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
6		sseq2 3960	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑥 ⊆ 𝑎 ↔ 𝑥 ⊆ 𝐴))
7		sseq2 3960	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑦 ⊆ 𝑏 ↔ 𝑦 ⊆ 𝐵))
8	6, 7	bi2anan9 638	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)))
9	8	anbi1d 631	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)))
10	9	opabbidv 5164	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)})
11	10	adantl 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)})
12	5, 11	mpteq12dv 5185	. 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 3464	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
15		bj-imdirvallem.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
16	15	elexd 3464	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
17	13, 15	xpexd 7696	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
18	17	pwexd 5324	. . 3 ⊢ (𝜑 → 𝒫 (𝐴 × 𝐵) ∈ V)
19	18	mptexd 7170	. 2 ⊢ (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}) ∈ V)
20	2, 12, 14, 16, 19	ovmpod 7510	1 ⊢ (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  𝒫 cpw 4554  {copab 5160   ↦ cmpt 5179   × cxp 5622  (class class class)co 7358   ∈ cmpo 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363

This theorem is referenced by:  bj-imdirval  37386  bj-iminvval  37398