Theorem fnse 8080

Description: Condition for the well-order in fnwe 8079 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	ffvelcdmda 7032	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵)
4		seex 5584	. . . . . . 7 ⊢ ((𝑅 Se 𝐵 ∧ (𝐹‘𝑧) ∈ 𝐵) → {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V)
5	1, 3, 4	syl2an2r 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V)
6		snex 5375	. . . . . 6 ⊢ {(𝐹‘𝑧)} ∈ V
7		unexg 7693	. . . . . 6 ⊢ (({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V ∧ {(𝐹‘𝑧)} ∈ V) → ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V)
8	5, 6, 7	sylancl 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V)
9		imaeq2 6015	. . . . . . . . 9 ⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → (◡𝐹 “ 𝑤) = (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
10	9	eleq1d 2825	. . . . . . . 8 ⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → ((◡𝐹 “ 𝑤) ∈ V ↔ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V))
11	10	imbi2d 341	. . . . . . 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 3502	. . . . . 6 ⊢ (({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V → (𝜑 → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V))
14	13	impcom 408	. . . . 5 ⊢ ((𝜑 ∧ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V) → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V)
15	8, 14	syldan 597	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V)
16		inss2 4173	. . . . . 6 ⊢ (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝑇 “ {𝑧})
17		vex 3436	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
18	17	eliniseg 6053	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (𝑤 ∈ (◡𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧))
19	18	elv 3437	. . . . . . . 8 ⊢ (𝑤 ∈ (◡𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧)
20		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
21		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
22	20, 21	breqan12d 5095	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
23	20, 21	eqeqan12d 2754	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑤) = (𝐹‘𝑧)))
24		breq12 5084	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥𝑆𝑦 ↔ 𝑤𝑆𝑧))
25	23, 24	anbi12d 638	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦) ↔ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)))
26	22, 25	orbi12d 924	. . . . . . . . . 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 5718	. . . . . . . . 9 ⊢ (𝑤𝑇𝑧 ↔ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧))))
29	2	ffvelcdmda 7032	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐵)
30	29	adantrr 723	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑤) ∈ 𝐵)
31		breq1 5082	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = (𝐹‘𝑤) → (𝑢𝑅(𝐹‘𝑧) ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
32	31	elrab3 3637	. . . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑤)𝑅(𝐹‘𝑧) → (𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)}))
35		fvex 6847	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑤) ∈ V
36	35	elsn 4577	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑤) ∈ {(𝐹‘𝑧)} ↔ (𝐹‘𝑤) = (𝐹‘𝑧))
37	36	biranri 506	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧) → (𝐹‘𝑤) ∈ {(𝐹‘𝑧)})
38	37	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧) → (𝐹‘𝑤) ∈ {(𝐹‘𝑧)}))
39	34, 38	orim12d 972	. . . . . . . . . . . . 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)})))
40		elun 4090	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ↔ ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∨ (𝐹‘𝑤) ∈ {(𝐹‘𝑧)}))
41	39, 40	imbitrrdi 253	. . . . . . . . . . . 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)})))
42		simprl 776	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑤 ∈ 𝐴)
43	41, 42	jctild 530	. . . . . . . . . . 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)}))))
44	2	ffnd 6663	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn 𝐴)
45	44	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐹 Fn 𝐴)
46		elpreima 7006	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
47	45, 46	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)}))))
48	43, 47	sylibrd 260	. . . . . . . . . 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)}))))
49	48	expimpd 454	. . . . . . . . 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)}))))
50	28, 49	biimtrid 243	. . . . . . . 8 ⊢ (𝜑 → (𝑤𝑇𝑧 → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
51	19, 50	biimtrid 243	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ (◡𝑇 “ {𝑧}) → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
52	51	ssrdv 3928	. . . . . 6 ⊢ (𝜑 → (◡𝑇 “ {𝑧}) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
53	16, 52	sstrid 3933	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
54	53	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
55	15, 54	ssexd 5259	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V)
56	55	ralrimiva 3132	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V)
57		dfse2 6059	. 2 ⊢ (𝑇 Se 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V)
58	56, 57	sylibr 235	1 ⊢ (𝜑 → 𝑇 Se 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {crab 3392  Vcvv 3432   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  {csn 4562   class class class wbr 5079  {copab 5141   Se wse 5576  ^◡ccnv 5624   “ cima 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-se 5579  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  r0weon  9932