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Theorem f1oweALT 7978

Description: Alternate proof of f1owe 7357, more direct since not using the isomorphism predicate, but requiring ax-un 7738. (Contributed by NM, 4-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
f1oweALT.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}

**Assertion**
Ref	Expression
f1oweALT	⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))

Distinct variable groups: 𝑥,𝑦,𝑆 𝑥,𝐹,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑦) 𝐵(𝑥,𝑦) 𝑅(𝑥,𝑦)

Proof of Theorem f1oweALT Dummy variables 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1ofo 6842	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
2		df-fo 6552	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
3		freq2 5645	. . . . . . 7 ⊢ (ran 𝐹 = 𝐵 → (𝑆 Fr ran 𝐹 ↔ 𝑆 Fr 𝐵))
4	3	biimprd 247	. . . . . 6 ⊢ (ran 𝐹 = 𝐵 → (𝑆 Fr 𝐵 → 𝑆 Fr ran 𝐹))
5		df-fn 6549	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6		df-fr 5629	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 Fr ran 𝐹 ↔ ∀𝑤((𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅) → ∃𝑢 ∈ 𝑤 ∀𝑓 ∈ 𝑤 ¬ 𝑓𝑆𝑢))
7		vex 3466	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑧 ∈ V
8	7	funimaex 6639	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun 𝐹 → (𝐹 “ 𝑧) ∈ V)
9		n0 4346	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝑧)
10		funfvima2 7240	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → (𝑤 ∈ 𝑧 → (𝐹‘𝑤) ∈ (𝐹 “ 𝑧)))
11		ne0i 4334	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐹‘𝑤) ∈ (𝐹 “ 𝑧) → (𝐹 “ 𝑧) ≠ ∅)
12	10, 11	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → (𝑤 ∈ 𝑧 → (𝐹 “ 𝑧) ≠ ∅))
13	12	exlimdv 1929	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → (∃𝑤 𝑤 ∈ 𝑧 → (𝐹 “ 𝑧) ≠ ∅))
14	9, 13	biimtrid 241	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → (𝑧 ≠ ∅ → (𝐹 “ 𝑧) ≠ ∅))
15	14	imp 405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → (𝐹 “ 𝑧) ≠ ∅)
16		imassrn 6072	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 “ 𝑧) ⊆ ran 𝐹
17	15, 16	jctil 518	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ((𝐹 “ 𝑧) ⊆ ran 𝐹 ∧ (𝐹 “ 𝑧) ≠ ∅))
18		sseq1 4004	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝐹 “ 𝑧) → (𝑤 ⊆ ran 𝐹 ↔ (𝐹 “ 𝑧) ⊆ ran 𝐹))
19		neeq1 2993	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝐹 “ 𝑧) → (𝑤 ≠ ∅ ↔ (𝐹 “ 𝑧) ≠ ∅))
20	18, 19	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = (𝐹 “ 𝑧) → ((𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅) ↔ ((𝐹 “ 𝑧) ⊆ ran 𝐹 ∧ (𝐹 “ 𝑧) ≠ ∅)))
21		raleq 3312	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝐹 “ 𝑧) → (∀𝑓 ∈ 𝑤 ¬ 𝑓𝑆𝑢 ↔ ∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢))
22	21	rexeqbi1dv 3324	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = (𝐹 “ 𝑧) → (∃𝑢 ∈ 𝑤 ∀𝑓 ∈ 𝑤 ¬ 𝑓𝑆𝑢 ↔ ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢))
23	20, 22	imbi12d 343	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (𝐹 “ 𝑧) → (((𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅) → ∃𝑢 ∈ 𝑤 ∀𝑓 ∈ 𝑤 ¬ 𝑓𝑆𝑢) ↔ (((𝐹 “ 𝑧) ⊆ ran 𝐹 ∧ (𝐹 “ 𝑧) ≠ ∅) → ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢)))
24	23	spcgv 3581	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 “ 𝑧) ∈ V → (∀𝑤((𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅) → ∃𝑢 ∈ 𝑤 ∀𝑓 ∈ 𝑤 ¬ 𝑓𝑆𝑢) → (((𝐹 “ 𝑧) ⊆ ran 𝐹 ∧ (𝐹 “ 𝑧) ≠ ∅) → ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢)))
25	17, 24	syl7 74	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 “ 𝑧) ∈ V → (∀𝑤((𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅) → ∃𝑢 ∈ 𝑤 ∀𝑓 ∈ 𝑤 ¬ 𝑓𝑆𝑢) → (((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢)))
26	8, 25	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Fun 𝐹 → (∀𝑤((𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅) → ∃𝑢 ∈ 𝑤 ∀𝑓 ∈ 𝑤 ¬ 𝑓𝑆𝑢) → (((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢)))
27	6, 26	biimtrid 241	. . . . . . . . . . . . . . . . . . 19 ⊢ (Fun 𝐹 → (𝑆 Fr ran 𝐹 → (((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢)))
28	27	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun 𝐹 → (((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → (𝑆 Fr ran 𝐹 → ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢)))
29	28	expd 414	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐹 → ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → (𝑧 ≠ ∅ → (𝑆 Fr ran 𝐹 → ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢))))
30	29	anabsi5 667	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → (𝑧 ≠ ∅ → (𝑆 Fr ran 𝐹 → ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢)))
31	30	impd 409	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → ((𝑧 ≠ ∅ ∧ 𝑆 Fr ran 𝐹) → ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢))
32		fores 6817	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → (𝐹 ↾ 𝑧):𝑧–onto→(𝐹 “ 𝑧))
33		vex 3466	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑣 ∈ V
34		vex 3466	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑤 ∈ V
35		fveq2 6893	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑣 → (𝐹‘𝑥) = (𝐹‘𝑣))
36	35	breq1d 5155	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑣 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑦)))
37		fveq2 6893	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
38	37	breq2d 5157	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑣)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑤)))
39		f1oweALT.1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}
40	33, 34, 36, 38, 39	brab 5541	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣𝑅𝑤 ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑤))
41		fvres 6912	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝑧 → ((𝐹 ↾ 𝑧)‘𝑣) = (𝐹‘𝑣))
42		fvres 6912	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ 𝑧 → ((𝐹 ↾ 𝑧)‘𝑤) = (𝐹‘𝑤))
43	41, 42	breqan12rd 5162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧) → (((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑤)))
44	40, 43	bitr4id 289	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧) → (𝑣𝑅𝑤 ↔ ((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤)))
45	44	notbid 317	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧) → (¬ 𝑣𝑅𝑤 ↔ ¬ ((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤)))
46	45	ralbidva 3166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝑧 → (∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤 ↔ ∀𝑣 ∈ 𝑧 ¬ ((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤)))
47	46	rexbiia 3082	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤 ↔ ∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ ((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤))
48		breq1 5148	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ↾ 𝑧)‘𝑣) = 𝑓 → (((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ 𝑓𝑆((𝐹 ↾ 𝑧)‘𝑤)))
49	48	notbid 317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ↾ 𝑧)‘𝑣) = 𝑓 → (¬ ((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ ¬ 𝑓𝑆((𝐹 ↾ 𝑧)‘𝑤)))
50	49	cbvfo 7295	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ↾ 𝑧):𝑧–onto→(𝐹 “ 𝑧) → (∀𝑣 ∈ 𝑧 ¬ ((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ ∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆((𝐹 ↾ 𝑧)‘𝑤)))
51	50	rexbidv 3169	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ↾ 𝑧):𝑧–onto→(𝐹 “ 𝑧) → (∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ ((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ ∃𝑤 ∈ 𝑧 ∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆((𝐹 ↾ 𝑧)‘𝑤)))
52		breq2 5149	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ↾ 𝑧)‘𝑤) = 𝑢 → (𝑓𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ 𝑓𝑆𝑢))
53	52	notbid 317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ↾ 𝑧)‘𝑤) = 𝑢 → (¬ 𝑓𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ ¬ 𝑓𝑆𝑢))
54	53	ralbidv 3168	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ↾ 𝑧)‘𝑤) = 𝑢 → (∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ ∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢))
55	54	cbvexfo 7296	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ↾ 𝑧):𝑧–onto→(𝐹 “ 𝑧) → (∃𝑤 ∈ 𝑧 ∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢))
56	51, 55	bitrd 278	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ↾ 𝑧):𝑧–onto→(𝐹 “ 𝑧) → (∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ ((𝐹 ↾ 𝑧)‘𝑣)𝑆((𝐹 ↾ 𝑧)‘𝑤) ↔ ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢))
57	47, 56	bitrid 282	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ↾ 𝑧):𝑧–onto→(𝐹 “ 𝑧) → (∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤 ↔ ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢))
58	32, 57	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → (∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤 ↔ ∃𝑢 ∈ (𝐹 “ 𝑧)∀𝑓 ∈ (𝐹 “ 𝑧) ¬ 𝑓𝑆𝑢))
59	31, 58	sylibrd 258	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹) → ((𝑧 ≠ ∅ ∧ 𝑆 Fr ran 𝐹) → ∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤))
60	59	exp4b 429	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝑧 ⊆ dom 𝐹 → (𝑧 ≠ ∅ → (𝑆 Fr ran 𝐹 → ∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤))))
61	60	com34 91	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝑧 ⊆ dom 𝐹 → (𝑆 Fr ran 𝐹 → (𝑧 ≠ ∅ → ∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤))))
62	61	com23 86	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑆 Fr ran 𝐹 → (𝑧 ⊆ dom 𝐹 → (𝑧 ≠ ∅ → ∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤))))
63	62	imp4a 421	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (𝑆 Fr ran 𝐹 → ((𝑧 ⊆ dom 𝐹 ∧ 𝑧 ≠ ∅) → ∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤)))
64	63	alrimdv 1925	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝑆 Fr ran 𝐹 → ∀𝑧((𝑧 ⊆ dom 𝐹 ∧ 𝑧 ≠ ∅) → ∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤)))
65		df-fr 5629	. . . . . . . . 9 ⊢ (𝑅 Fr dom 𝐹 ↔ ∀𝑧((𝑧 ⊆ dom 𝐹 ∧ 𝑧 ≠ ∅) → ∃𝑤 ∈ 𝑧 ∀𝑣 ∈ 𝑧 ¬ 𝑣𝑅𝑤))
66	64, 65	imbitrrdi 251	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝑆 Fr ran 𝐹 → 𝑅 Fr dom 𝐹))
67		freq2 5645	. . . . . . . . 9 ⊢ (dom 𝐹 = 𝐴 → (𝑅 Fr dom 𝐹 ↔ 𝑅 Fr 𝐴))
68	67	biimpd 228	. . . . . . . 8 ⊢ (dom 𝐹 = 𝐴 → (𝑅 Fr dom 𝐹 → 𝑅 Fr 𝐴))
69	66, 68	sylan9 506	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝑆 Fr ran 𝐹 → 𝑅 Fr 𝐴))
70	5, 69	sylbi 216	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑆 Fr ran 𝐹 → 𝑅 Fr 𝐴))
71	4, 70	sylan9r 507	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
72	2, 71	sylbi 216	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
73	1, 72	syl 17	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
74		df-f1o 6553	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
75		fveq2 6893	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
76	75	breq1d 5155	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑦)))
77		fveq2 6893	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
78	77	breq2d 5157	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑤)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑣)))
79	34, 33, 76, 78, 39	brab 5541	. . . . . . . . 9 ⊢ (𝑤𝑅𝑣 ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑣))
80	79	a1i 11	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑤𝑅𝑣 ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑣)))
81		f1fveq 7269	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑤) = (𝐹‘𝑣) ↔ 𝑤 = 𝑣))
82	81	bicomd 222	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑤 = 𝑣 ↔ (𝐹‘𝑤) = (𝐹‘𝑣)))
83	40	a1i 11	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑣𝑅𝑤 ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑤)))
84	80, 82, 83	3orbi123d 1432	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝑤𝑅𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣𝑅𝑤) ↔ ((𝐹‘𝑤)𝑆(𝐹‘𝑣) ∨ (𝐹‘𝑤) = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆(𝐹‘𝑤))))
85	84	2ralbidva 3207	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑤𝑅𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣𝑅𝑤) ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤)𝑆(𝐹‘𝑣) ∨ (𝐹‘𝑤) = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆(𝐹‘𝑤))))
86		breq1 5148	. . . . . . . . . 10 ⊢ ((𝐹‘𝑤) = 𝑢 → ((𝐹‘𝑤)𝑆(𝐹‘𝑣) ↔ 𝑢𝑆(𝐹‘𝑣)))
87		eqeq1 2730	. . . . . . . . . 10 ⊢ ((𝐹‘𝑤) = 𝑢 → ((𝐹‘𝑤) = (𝐹‘𝑣) ↔ 𝑢 = (𝐹‘𝑣)))
88		breq2 5149	. . . . . . . . . 10 ⊢ ((𝐹‘𝑤) = 𝑢 → ((𝐹‘𝑣)𝑆(𝐹‘𝑤) ↔ (𝐹‘𝑣)𝑆𝑢))
89	86, 87, 88	3orbi123d 1432	. . . . . . . . 9 ⊢ ((𝐹‘𝑤) = 𝑢 → (((𝐹‘𝑤)𝑆(𝐹‘𝑣) ∨ (𝐹‘𝑤) = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆(𝐹‘𝑤)) ↔ (𝑢𝑆(𝐹‘𝑣) ∨ 𝑢 = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆𝑢)))
90	89	ralbidv 3168	. . . . . . . 8 ⊢ ((𝐹‘𝑤) = 𝑢 → (∀𝑣 ∈ 𝐴 ((𝐹‘𝑤)𝑆(𝐹‘𝑣) ∨ (𝐹‘𝑤) = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆(𝐹‘𝑤)) ↔ ∀𝑣 ∈ 𝐴 (𝑢𝑆(𝐹‘𝑣) ∨ 𝑢 = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆𝑢)))
91	90	cbvfo 7295	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤)𝑆(𝐹‘𝑣) ∨ (𝐹‘𝑤) = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆(𝐹‘𝑤)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐴 (𝑢𝑆(𝐹‘𝑣) ∨ 𝑢 = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆𝑢)))
92		breq2 5149	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣) = 𝑓 → (𝑢𝑆(𝐹‘𝑣) ↔ 𝑢𝑆𝑓))
93		eqeq2 2738	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣) = 𝑓 → (𝑢 = (𝐹‘𝑣) ↔ 𝑢 = 𝑓))
94		breq1 5148	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣) = 𝑓 → ((𝐹‘𝑣)𝑆𝑢 ↔ 𝑓𝑆𝑢))
95	92, 93, 94	3orbi123d 1432	. . . . . . . . 9 ⊢ ((𝐹‘𝑣) = 𝑓 → ((𝑢𝑆(𝐹‘𝑣) ∨ 𝑢 = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆𝑢) ↔ (𝑢𝑆𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓𝑆𝑢)))
96	95	cbvfo 7295	. . . . . . . 8 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑣 ∈ 𝐴 (𝑢𝑆(𝐹‘𝑣) ∨ 𝑢 = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆𝑢) ↔ ∀𝑓 ∈ 𝐵 (𝑢𝑆𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓𝑆𝑢)))
97	96	ralbidv 3168	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐴 (𝑢𝑆(𝐹‘𝑣) ∨ 𝑢 = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆𝑢) ↔ ∀𝑢 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑢𝑆𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓𝑆𝑢)))
98	91, 97	bitrd 278	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤)𝑆(𝐹‘𝑣) ∨ (𝐹‘𝑤) = (𝐹‘𝑣) ∨ (𝐹‘𝑣)𝑆(𝐹‘𝑤)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑢𝑆𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓𝑆𝑢)))
99	85, 98	sylan9bb 508	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) → (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑤𝑅𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣𝑅𝑤) ↔ ∀𝑢 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑢𝑆𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓𝑆𝑢)))
100	74, 99	sylbi 216	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑤𝑅𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣𝑅𝑤) ↔ ∀𝑢 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑢𝑆𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓𝑆𝑢)))
101	100	biimprd 247	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (∀𝑢 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑢𝑆𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓𝑆𝑢) → ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑤𝑅𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣𝑅𝑤)))
102	73, 101	anim12d 607	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ((𝑆 Fr 𝐵 ∧ ∀𝑢 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑢𝑆𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓𝑆𝑢)) → (𝑅 Fr 𝐴 ∧ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑤𝑅𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣𝑅𝑤))))
103		dfwe2 7774	. 2 ⊢ (𝑆 We 𝐵 ↔ (𝑆 Fr 𝐵 ∧ ∀𝑢 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑢𝑆𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓𝑆𝑢)))
104		dfwe2 7774	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑤𝑅𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣𝑅𝑤)))
105	102, 103, 104	3imtr4g 295	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ w3o 1083 ∀wal 1532 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 Vcvv 3462 ⊆ wss 3946 ∅c0 4322 class class class wbr 5145 {copab 5207 Fr wfr 5626 We wwe 5628 dom cdm 5674 ran crn 5675 ↾ cres 5676 “ cima 5677 Fun wfun 6540 Fn wfn 6541 –1-1→wf1 6543 –onto→wfo 6544 –1-1-onto→wf1o 6545 ‘cfv 6546

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554

This theorem is referenced by: (None)

W3C validator