Theorem bj-imdirval3 37172

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 37171	. . . . 5 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
5	4	breqd 5118	. . . 4 ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ 𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}𝑌))
6		brabv 5528	. . . 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 3978	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
16		simpr 484	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
17	16	sseq1d 3978	. . . . . . 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 6027	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑅 “ 𝑥) = (𝑅 “ 𝑋))
20		id 22	. . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌)
21	19, 20	eqeqan12d 2743	. . . . . 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 37136	. . 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 5279	. . . . . . 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 5279	. . . . . . 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 1540   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914   class class class wbr 5107  {copab 5169   × cxp 5636   “ cima 5641  ‘cfv 6511  (class class class)co 7387  𝒫_*cimdir 37166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-imdir 37167

This theorem is referenced by: (None)