Theorem fnse 8117

Description: Condition for the well-order in fnwe 8116 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 7069	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵)
4		seex 5611	. . . . . . 7 ⊢ ((𝑅 Se 𝐵 ∧ (𝐹‘𝑧) ∈ 𝐵) → {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V)
5	1, 3, 4	syl2an2r 697	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V)
6		snex 5401	. . . . . 6 ⊢ {(𝐹‘𝑧)} ∈ V
7		unexg 7730	. . . . . 6 ⊢ (({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V ∧ {(𝐹‘𝑧)} ∈ V) → ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V)
8	5, 6, 7	sylancl 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V)
9		imaeq2 6049	. . . . . . . . 9 ⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → (◡𝐹 “ 𝑤) = (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
10	9	eleq1d 2850	. . . . . . . 8 ⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → ((◡𝐹 “ 𝑤) ∈ V ↔ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V))
11	10	imbi2d 343	. . . . . . 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 3525	. . . . . 6 ⊢ (({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V → (𝜑 → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V))
14	13	impcom 412	. . . . 5 ⊢ ((𝜑 ∧ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V) → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V)
15	8, 14	syldan 602	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V)
16		inss2 4192	. . . . . 6 ⊢ (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝑇 “ {𝑧})
17		vex 3461	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
18	17	eliniseg 6087	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (𝑤 ∈ (◡𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧))
19	18	elv 3462	. . . . . . . 8 ⊢ (𝑤 ∈ (◡𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧)
20		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
21		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
22	20, 21	breqan12d 5121	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
23	20, 21	eqeqan12d 2779	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑤) = (𝐹‘𝑧)))
24		breq12 5110	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥𝑆𝑦 ↔ 𝑤𝑆𝑧))
25	23, 24	anbi12d 643	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦) ↔ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)))
26	22, 25	orbi12d 931	. . . . . . . . . 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 5745	. . . . . . . . 9 ⊢ (𝑤𝑇𝑧 ↔ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧))))
29	2	ffvelcdmda 7069	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐵)
30	29	adantrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑤) ∈ 𝐵)
31		breq1 5108	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = (𝐹‘𝑤) → (𝑢𝑅(𝐹‘𝑧) ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
32	31	elrab3 3654	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑤) ∈ 𝐵 → ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
33	30, 32	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧)))
34	33	biimprd 251	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑤)𝑅(𝐹‘𝑧) → (𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)}))
35		fvex 6884	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑤) ∈ V
36	35	elsn 4600	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑤) ∈ {(𝐹‘𝑧)} ↔ (𝐹‘𝑤) = (𝐹‘𝑧))
37	36	biranri 510	. . . . . . . . . . . . . . 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 979	. . . . . . . . . . . . 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 4109	. . . . . . . . . . . . 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 255	. . . . . . . . . . . 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 782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑤 ∈ 𝐴)
43	41, 42	jctild 534	. . . . . . . . . . 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 6696	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn 𝐴)
45	44	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐹 Fn 𝐴)
46		elpreima 7043	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 262	. . . . . . . . . 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 458	. . . . . . . . 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 245	. . . . . . . 8 ⊢ (𝜑 → (𝑤𝑇𝑧 → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
51	19, 50	biimtrid 245	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ (◡𝑇 “ {𝑧}) → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))))
52	51	ssrdv 3945	. . . . . 6 ⊢ (𝜑 → (◡𝑇 “ {𝑧}) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
53	16, 52	sstrid 3950	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
54	53	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))
55	15, 54	ssexd 5285	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V)
56	55	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V)
57		dfse2 6093	. 2 ⊢ (𝑇 Se 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V)
58	56, 57	sylibr 237	1 ⊢ (𝜑 → 𝑇 Se 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417  Vcvv 3457   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  {csn 4585   class class class wbr 5105  {copab 5167   Se wse 5603  ^◡ccnv 5651   “ cima 5655   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-se 5606  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533

This theorem is referenced by:  r0weon  9984