Theorem bj-imdirval3 38085

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 38084	. . . . 5 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
5	4	breqd 5114	. . . 4 ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ 𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}𝑌))
6		brabv 5541	. . . 4 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
7	5, 6	biimtrdi 256	. . 3 ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
8	7	pm4.71rd 572	. 2 ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌)))
9		simpl 488	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
10	9	adantl 487	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → 𝑋 ∈ V)
11		simpr 490	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
12	11	adantl 487	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → 𝑌 ∈ V)
13	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
14		simpl 488	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
15	14	sseq1d 3962	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
16		simpr 490	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
17	16	sseq1d 3962	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑦 ⊆ 𝐵 ↔ 𝑌 ⊆ 𝐵))
18	15, 17	anbi12d 644	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵)))
19		imaeq2 6048	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑅 “ 𝑥) = (𝑅 “ 𝑋))
20		id 23	. . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌)
21	19, 20	eqeqan12d 2775	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑅 “ 𝑥) = 𝑦 ↔ (𝑅 “ 𝑋) = 𝑌))
22	18, 21	anbi12d 644	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))
23	22	adantl 487	. . . 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 38049	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V)) → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))
25	24	pm5.32da 590	. 2 ⊢ (𝜑 → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌))))
26		simpr 490	. . 3 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)) → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌))
27	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝐴) → 𝐴 ∈ 𝑈)
28		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
29	27, 28	ssexd 5286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ V)
30	29	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑋 ⊆ 𝐴 → 𝑋 ∈ V))
31	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ⊆ 𝐵) → 𝐵 ∈ 𝑉)
32		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ⊆ 𝐵) → 𝑌 ⊆ 𝐵)
33	31, 32	ssexd 5286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ⊆ 𝐵) → 𝑌 ∈ V)
34	33	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑌 ⊆ 𝐵 → 𝑌 ∈ V))
35	30, 34	anim12d 621	. . . . 5 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
36	35	adantrd 497	. . . 4 ⊢ (𝜑 → (((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
37	36	ancrd 561	. . 3 ⊢ (𝜑 → (((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌))))
38	26, 37	impbid2 229	. 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 308	1 ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899   class class class wbr 5103  {copab 5167   × cxp 5649   “ cima 5654  ‘cfv 6537  (class class class)co 7418  𝒫_*cimdir 38079

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-imdir 38080

This theorem is used by: (None)