Theorem fnse 7945

Description: Condition for the well-order in fnwe 7944 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)

Hypotheses

Ref	Expression
fnse.1	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}
fnse.2	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fnse.3	⊢ (𝜑 → 𝑅 Se 𝐵)
fnse.4	⊢ (𝜑 → (◡𝐹 “ 𝑤) ∈ V)

Assertion

Ref	Expression
fnse	⊢ (𝜑 → 𝑇 Se 𝐴)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑤,𝐵   𝑥,𝑤,𝑦,𝐹   𝜑,𝑤   𝑤,𝑅,𝑥,𝑦   𝑥,𝑆,𝑦   𝑤,𝑇

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑤)   𝐵(𝑥,𝑦)   𝑆(𝑤)   𝑇(𝑥,𝑦)

Proof of Theorem fnse

Dummy variables 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnse.3	. . . . . . 7 ⊢ (𝜑 → 𝑅 Se 𝐵)
2		fnse.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	ffvelrnda 6943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵)
4		seex 5542	. . . . . . 7 ⊢ ((𝑅 Se 𝐵 ∧ (𝐹‘𝑧) ∈ 𝐵) → {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V)
5	1, 3, 4	syl2an2r 681	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V)
6		snex 5349	. . . . . 6 ⊢ {(𝐹‘𝑧)} ∈ V
7		unexg 7577	. . . . . 6 ⊢ (({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V ∧ {(𝐹‘𝑧)} ∈ V) → ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V)
8	5, 6, 7	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V)
9		imaeq2 5954	. . . . . . . . 9 ⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → (◡𝐹 “ 𝑤) = (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
10	9	eleq1d 2823	. . . . . . . 8 ⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → ((◡𝐹 “ 𝑤) ∈ V ↔ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V))
11	10	imbi2d 340	. . . . . . 7 ⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → ((𝜑 → (◡𝐹 “ 𝑤) ∈ V) ↔ (𝜑 → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V)))
12		fnse.4	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑤) ∈ V)
13	11, 12	vtoclg 3495	. . . . . 6 ⊢ (({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V → (𝜑 → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V))
14	13	impcom 407	. . . . 5 ⊢ ((𝜑 ∧ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V) → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V)
15	8, 14	syldan 590	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V)
16		inss2 4160	. . . . . 6 ⊢ (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝑇 “ {𝑧})
17		vex 3426	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
18	17	eliniseg 5991	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (𝑤 ∈ (◡𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧))
19	18	elv 3428	. . . . . . . 8 ⊢ (𝑤 ∈ (◡𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧)
20		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
21		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
22	20, 21	breqan12d 5086	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
23	20, 21	eqeqan12d 2752	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑤) = (𝐹‘𝑧)))
24		breq12 5075	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥𝑆𝑦 ↔ 𝑤𝑆𝑧))
25	23, 24	anbi12d 630	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦) ↔ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)))
26	22, 25	orbi12d 915	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧))))
27		fnse.1	. . . . . . . . . 10 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}
28	26, 27	brab2a 5670	. . . . . . . . 9 ⊢ (𝑤𝑇𝑧 ↔ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧))))
29	2	ffvelrnda 6943	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐵)
30	29	adantrr 713	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑤) ∈ 𝐵)
31		breq1 5073	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = (𝐹‘𝑤) → (𝑢𝑅(𝐹‘𝑧) ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
32	31	elrab3 3618	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑤) ∈ 𝐵 → ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
33	30, 32	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
34	33	biimprd 247	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑤)𝑅(𝐹‘𝑧) → (𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)}))
35		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧) → (𝐹‘𝑤) = (𝐹‘𝑧))
36		fvex 6769	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑤) ∈ V
37	36	elsn 4573	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑤) ∈ {(𝐹‘𝑧)} ↔ (𝐹‘𝑤) = (𝐹‘𝑧))
38	35, 37	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧) → (𝐹‘𝑤) ∈ {(𝐹‘𝑧)})
39	38	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧) → (𝐹‘𝑤) ∈ {(𝐹‘𝑧)}))
40	34, 39	orim12d 961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)) → ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∨ (𝐹‘𝑤) ∈ {(𝐹‘𝑧)})))
41		elun 4079	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ↔ ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∨ (𝐹‘𝑤) ∈ {(𝐹‘𝑧)}))
42	40, 41	syl6ibr 251	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)) → (𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
43		simprl 767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑤 ∈ 𝐴)
44	42, 43	jctild 525	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)) → (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
45	2	ffnd 6585	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn 𝐴)
46	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐹 Fn 𝐴)
47		elpreima 6917	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
48	46, 47	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
49	44, 48	sylibrd 258	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)) → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
50	49	expimpd 453	. . . . . . . . 9 ⊢ (𝜑 → (((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧))) → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
51	28, 50	syl5bi 241	. . . . . . . 8 ⊢ (𝜑 → (𝑤𝑇𝑧 → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
52	19, 51	syl5bi 241	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ (◡𝑇 “ {𝑧}) → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
53	52	ssrdv 3923	. . . . . 6 ⊢ (𝜑 → (◡𝑇 “ {𝑧}) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
54	16, 53	sstrid 3928	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
55	54	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
56	15, 55	ssexd 5243	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V)
57	56	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V)
58		dfse2 5997	. 2 ⊢ (𝑇 Se 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V)
59	57, 58	sylibr 233	1 ⊢ (𝜑 → 𝑇 Se 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  {csn 4558   class class class wbr 5070  {copab 5132   Se wse 5533  ^◡ccnv 5579   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-se 5536  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426

This theorem is referenced by:  r0weon  9699