Theorem fpwrelmapffslem 30644

Description: Lemma for fpwrelmapffs 30646. For this theorem, the sets 𝐴 and 𝐵 could be infinite, but the relation 𝑅 itself is finite. (Contributed by Thierry Arnoux, 1-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)

Hypotheses

Ref	Expression
fpwrelmapffslem.1	⊢ 𝐴 ∈ V
fpwrelmapffslem.2	⊢ 𝐵 ∈ V
fpwrelmapffslem.3	⊢ (𝜑 → 𝐹:𝐴⟶𝒫 𝐵)
fpwrelmapffslem.4	⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})

Assertion

Ref	Expression
fpwrelmapffslem	⊢ (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem fpwrelmapffslem

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

Step	Hyp	Ref	Expression
1		fpwrelmapffslem.4	. . 3 ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
2		relopabv 5665	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
3		releq 5622	. . . 4 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}))
4	2, 3	mpbiri 261	. . 3 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → Rel 𝑅)
5		relfi 30517	. . 3 ⊢ (Rel 𝑅 → (𝑅 ∈ Fin ↔ (dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin)))
6	1, 4, 5	3syl 18	. 2 ⊢ (𝜑 → (𝑅 ∈ Fin ↔ (dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin)))
7		rexcom4 3163	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
8		ancom 464	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ↔ (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
9	8	exbii 1854	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
10		fvex 6689	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑥) ∈ V
11		eleq2 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ (𝐹‘𝑥)))
12	10, 11	ceqsexv 3445	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ↔ 𝑤 ∈ (𝐹‘𝑥))
13	9, 12	bitr3i 280	. . . . . . . . . . . . . 14 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ 𝑤 ∈ (𝐹‘𝑥))
14	13	rexbii 3161	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥))
15		r19.42v 3254	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
16	15	exbii 1854	. . . . . . . . . . . . 13 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
17	7, 14, 16	3bitr3ri 305	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥))
18		df-rex 3059	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)))
19	17, 18	bitr2i 279	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥))))
21		vex 3402	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
22		eleq1w 2815	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑤 ∈ (𝐹‘𝑥)))
23	22	anbi2d 632	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥))))
24	23	exbidv 1928	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥))))
25	21, 24	elab 3573	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)))
26		eluniab 4811	. . . . . . . . . 10 ⊢ (𝑤 ∈ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
27	20, 25, 26	3bitr4g 317	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↔ 𝑤 ∈ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)}))
28	27	eqrdv 2736	. . . . . . . 8 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)})
29	28	eleq1d 2817	. . . . . . 7 ⊢ (𝜑 → ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin ↔ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
30	29	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin ↔ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
31		fpwrelmapffslem.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝒫 𝐵)
32		ffn 6504	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝒫 𝐵 → 𝐹 Fn 𝐴)
33		fnrnfv 6731	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)})
34	31, 32, 33	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)})
35	34	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)})
36		0ex 5175	. . . . . . . . . . 11 ⊢ ∅ ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ∅ ∈ V)
38		fpwrelmapffslem.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ V
39		fex 7001	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝒫 𝐵 ∧ 𝐴 ∈ V) → 𝐹 ∈ V)
40	31, 38, 39	sylancl 589	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
41	40	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → 𝐹 ∈ V)
42	31	ffund 6508	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
43	42	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → Fun 𝐹)
44		opabdm 30527	. . . . . . . . . . . . . 14 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → dom 𝑅 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
45	1, 44	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝑅 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
46	38, 39	mpan2 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝒫 𝐵 → 𝐹 ∈ V)
47		suppimacnv 7871	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
48	36, 47	mpan2 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ V → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
49	31, 46, 48	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
50	31	feqmptd 6739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
51	50	cnveqd 5718	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
52	51	imaeq1d 5902	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {∅})) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})))
53	49, 52	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp ∅) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})))
54		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
55	54	mptpreima 6070	. . . . . . . . . . . . . . 15 ⊢ (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})}
56	53, 55	eqtrdi 2789	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})})
57		suppvalfn 7866	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅})
58	38, 36, 57	mp3an23 1454	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn 𝐴 → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅})
59	31, 32, 58	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅})
60		n0 4235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
61	60	rabbii 3374	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)}
62	61	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)})
63	59, 56, 62	3eqtr3d 2781	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)})
64		df-rab 3062	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
65		19.42v 1961	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)))
66	65	abbii 2803	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
67	64, 66	eqtr4i 2764	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
68	67	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
69	56, 63, 68	3eqtrd 2777	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
70	45, 69	eqtr4d 2776	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑅 = (𝐹 supp ∅))
71	70	eleq1d 2817	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝑅 ∈ Fin ↔ (𝐹 supp ∅) ∈ Fin))
72	71	biimpa 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (𝐹 supp ∅) ∈ Fin)
73	37, 41, 43, 72	ffsrn 30641	. . . . . . . . 9 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 ∈ Fin)
74	35, 73	eqeltrrd 2834	. . . . . . . 8 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin)
75		unifi 8888	. . . . . . . . 9 ⊢ (({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin)
76	75	ex 416	. . . . . . . 8 ⊢ ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
77	74, 76	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
78		unifi3 30624	. . . . . . 7 ⊢ (∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin)
79	77, 78	impbid1 228	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin ↔ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
80	30, 79	bitr4d 285	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin))
81		opabrn 30528	. . . . . . . 8 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
82	1, 81	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
83	82	eleq1d 2817	. . . . . 6 ⊢ (𝜑 → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin))
84	83	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin))
85	35	sseq1d 3908	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝐹 ⊆ Fin ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin))
86	80, 84, 85	3bitr4d 314	. . . 4 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ ran 𝐹 ⊆ Fin))
87	86	pm5.32da 582	. . 3 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ (dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
88	71	anbi1d 633	. . 3 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
89	87, 88	bitrd 282	. 2 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
90		ancom 464	. . 3 ⊢ (((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin))
91	90	a1i 11	. 2 ⊢ (𝜑 → (((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
92	6, 89, 91	3bitrd 308	1 ⊢ (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542  ∃wex 1786   ∈ wcel 2114  {cab 2716   ≠ wne 2934  ∃wrex 3054  {crab 3057  Vcvv 3398   ∖ cdif 3840   ⊆ wss 3843  ∅c0 4211  𝒫 cpw 4488  {csn 4516  ∪ cuni 4796  {copab 5092   ↦ cmpt 5110  ^◡ccnv 5524  dom cdm 5525  ran crn 5526   “ cima 5528  Rel wrel 5530  Fun wfun 6333   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  (class class class)co 7172   supp csupp 7858  Fincfn 8557

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-ac2 9965

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-om 7602  df-1st 7716  df-2nd 7717  df-supp 7859  df-wrecs 7978  df-recs 8039  df-1o 8133  df-er 8322  df-map 8441  df-en 8558  df-dom 8559  df-fin 8561  df-card 9443  df-acn 9446  df-ac 9618

This theorem is referenced by:  fpwrelmapffs  30646