Theorem bj-imdirval3 37389

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

Hypotheses

Ref	Expression
bj-imdirval3.exa	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-imdirval3.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
bj-imdirval3.arg	⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))

Assertion

Ref	Expression
bj-imdirval3	⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))

Proof of Theorem bj-imdirval3

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-imdirval3.exa	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		bj-imdirval3.exb	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3		bj-imdirval3.arg	. . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
4	1, 2, 3	bj-imdirval2 37388	. . . . 5 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
5	4	breqd 5109	. . . 4 ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ 𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}𝑌))
6		brabv 5514	. . . 4 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
7	5, 6	biimtrdi 253	. . 3 ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
8	7	pm4.71rd 562	. 2 ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌)))
9		simpl 482	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
10	9	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → 𝑋 ∈ V)
11		simpr 484	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
12	11	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → 𝑌 ∈ V)
13	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
14		simpl 482	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
15	14	sseq1d 3965	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
16		simpr 484	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
17	16	sseq1d 3965	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑦 ⊆ 𝐵 ↔ 𝑌 ⊆ 𝐵))
18	15, 17	anbi12d 632	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵)))
19		imaeq2 6015	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑅 “ 𝑥) = (𝑅 “ 𝑋))
20		id 22	. . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌)
21	19, 20	eqeqan12d 2750	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑅 “ 𝑥) = 𝑦 ↔ (𝑅 “ 𝑋) = 𝑌))
22	18, 21	anbi12d 632	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))
23	22	adantl 481	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))
24	10, 12, 13, 23	brabd 37353	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))
25	24	pm5.32da 579	. 2 ⊢ (𝜑 → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌))))
26		simpr 484	. . 3 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)) → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌))
27	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝐴) → 𝐴 ∈ 𝑈)
28		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
29	27, 28	ssexd 5269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ V)
30	29	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑋 ⊆ 𝐴 → 𝑋 ∈ V))
31	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ⊆ 𝐵) → 𝐵 ∈ 𝑉)
32		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ⊆ 𝐵) → 𝑌 ⊆ 𝐵)
33	31, 32	ssexd 5269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ⊆ 𝐵) → 𝑌 ∈ V)
34	33	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑌 ⊆ 𝐵 → 𝑌 ∈ V))
35	30, 34	anim12d 609	. . . . 5 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
36	35	adantrd 491	. . . 4 ⊢ (𝜑 → (((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
37	36	ancrd 551	. . 3 ⊢ (𝜑 → (((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌))))
38	26, 37	impbid2 226	. 2 ⊢ (𝜑 → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)) ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))
39	8, 25, 38	3bitrd 305	1 ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901   class class class wbr 5098  {copab 5160   × cxp 5622   “ cima 5627  ‘cfv 6492  (class class class)co 7358  𝒫_*cimdir 37383

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  df-imdir 37384

This theorem is referenced by: (None)