Theorem bropfvvvv 7257

Description: If a binary relation holds for the result of an operation which is a function value, the involved classes are sets. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.)

Hypotheses

Ref	Expression
bropfvvvv.o	⊢ 𝑂 = (𝑎 ∈ 𝑈 ↦ (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))
bropfvvvv.oo	⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (𝐵(𝑂‘𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
bropfvvvv.s	⊢ (𝑎 = 𝐴 → 𝑉 = 𝑆)
bropfvvvv.t	⊢ (𝑎 = 𝐴 → 𝑊 = 𝑇)
bropfvvvv.p	⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bropfvvvv	⊢ ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))

Distinct variable groups:   𝑈,𝑎   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒   𝑆,𝑎,𝑏,𝑐   𝑇,𝑎,𝑏,𝑐   𝜓,𝑎

Allowed substitution hints:   𝜑(𝑒,𝑎,𝑏,𝑐,𝑑)   𝜓(𝑒,𝑏,𝑐,𝑑)   𝜃(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐷(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑆(𝑒,𝑑)   𝑇(𝑒,𝑑)   𝑈(𝑒,𝑏,𝑐,𝑑)   𝐸(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑂(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑉(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑊(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑋(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑌(𝑒,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem bropfvvvv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brovpreldm 7254	. 2 ⊢ (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → ⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴))
2		bropfvvvv.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝑎 ∈ 𝑈 ↦ (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))
3	2	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → 𝑂 = (𝑎 ∈ 𝑈 ↦ (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑})))
4		bropfvvvv.s	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → 𝑉 = 𝑆)
5		bropfvvvv.t	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → 𝑊 = 𝑇)
6		bropfvvvv.p	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))
7	6	opabbidv 4716	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → {⟨𝑑, 𝑒⟩ ∣ 𝜑} = {⟨𝑑, 𝑒⟩ ∣ 𝜓})
8	4, 5, 7	mpt2eq123dv 6717	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}) = (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}))
9	8	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) ∧ 𝑎 = 𝐴) → (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}) = (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}))
10		simpl 473	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → 𝐴 ∈ 𝑈)
11		simpr 477	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V)
12	3, 9, 10, 11	fvmptd 6288	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (𝑂‘𝐴) = (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}))
13	12	dmeqd 5326	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → dom (𝑂‘𝐴) = dom (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}))
14	13	eleq2d 2687	. . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ dom (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓})))
15		dmoprabss 6742	. . . . . . . . 9 ⊢ dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})} ⊆ (𝑆 × 𝑇)
16	15	sseli 3599	. . . . . . . 8 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})} → ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇))
17		bropfvvvv.oo	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (𝐵(𝑂‘𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
18	2, 17	bropfvvvvlem 7256	. . . . . . . . 9 ⊢ ((⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸) → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
19	18	ex 450	. . . . . . . 8 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
20	16, 19	syl 17	. . . . . . 7 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})} → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
21		df-mpt2 6655	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) = {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})}
22	21	dmeqi 5325	. . . . . . 7 ⊢ dom (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) = dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})}
23	20, 22	eleq2s 2719	. . . . . 6 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
24	14, 23	syl6bi 243	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
25	24	com23 86	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
26	25	a1d 25	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
27		ianor 509	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) ↔ (¬ 𝐴 ∈ 𝑈 ∨ ¬ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V))
28	2	fvmptndm 6308	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ 𝑈 → (𝑂‘𝐴) = ∅)
29	28	dmeqd 5326	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ 𝑈 → dom (𝑂‘𝐴) = dom ∅)
30	29	eleq2d 2687	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ 𝑈 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ dom ∅))
31		dm0 5339	. . . . . . . . . 10 ⊢ dom ∅ = ∅
32	31	eleq2i 2693	. . . . . . . . 9 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom ∅ ↔ ⟨𝐵, 𝐶⟩ ∈ ∅)
33	30, 32	syl6bb 276	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ 𝑈 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ∅))
34		noel 3919	. . . . . . . . 9 ⊢ ¬ ⟨𝐵, 𝐶⟩ ∈ ∅
35	34	pm2.21i 116	. . . . . . . 8 ⊢ (⟨𝐵, 𝐶⟩ ∈ ∅ → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
36	33, 35	syl6bi 243	. . . . . . 7 ⊢ (¬ 𝐴 ∈ 𝑈 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
37	36	a1d 25	. . . . . 6 ⊢ (¬ 𝐴 ∈ 𝑈 → ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
38		notnotb 304	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑈 ↔ ¬ ¬ 𝐴 ∈ 𝑈)
39		elex 3212	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ 𝑋 → 𝑆 ∈ V)
40		elex 3212	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ 𝑌 → 𝑇 ∈ V)
41	39, 40	anim12i 590	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
42	41	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
43		mpt2exga 7246	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V)
44	42, 43	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌)) → (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V)
45	44	pm2.24d 147	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑈 ∧ (𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌)) → (¬ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
46	45	ex 450	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑈 → ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (¬ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))))
47	46	com23 86	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑈 → (¬ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V → ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))))
48	38, 47	sylbir 225	. . . . . . 7 ⊢ (¬ ¬ 𝐴 ∈ 𝑈 → (¬ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V → ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))))
49	48	imp 445	. . . . . 6 ⊢ ((¬ ¬ 𝐴 ∈ 𝑈 ∧ ¬ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
50	37, 49	jaoi3 1011	. . . . 5 ⊢ ((¬ 𝐴 ∈ 𝑈 ∨ ¬ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
51	27, 50	sylbi 207	. . . 4 ⊢ (¬ (𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
52	51	com34 91	. . 3 ⊢ (¬ (𝐴 ∈ 𝑈 ∧ (𝑏 ∈ 𝑆, 𝑐 ∈ 𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
53	26, 52	pm2.61i 176	. 2 ⊢ ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂‘𝐴) → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
54	1, 53	mpdi 45	1 ⊢ ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  ⟨cop 4183   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ‘cfv 5888  (class class class)co 6650  {coprab 6651   ↦ cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  wlkonprop  26554  wksonproplem  26601