Theorem fpwrelmapffslem 32805

Description: Lemma for fpwrelmapffs 32807. 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 5777	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
3		releq 5733	. . . 4 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}))
4	2, 3	mpbiri 258	. . 3 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → Rel 𝑅)
5		relfi 32672	. . 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 3265	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
8		ancom 460	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ↔ (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
9	8	exbii 1850	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
10		fvex 6854	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑥) ∈ V
11		eleq2 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ (𝐹‘𝑥)))
12	10, 11	ceqsexv 3479	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ↔ 𝑤 ∈ (𝐹‘𝑥))
13	9, 12	bitr3i 277	. . . . . . . . . . . . . 14 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ 𝑤 ∈ (𝐹‘𝑥))
14	13	rexbii 3085	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥))
15		r19.42v 3170	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
16	15	exbii 1850	. . . . . . . . . . . . 13 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
17	7, 14, 16	3bitr3ri 302	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥))
18		df-rex 3063	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)))
19	17, 18	bitr2i 276	. . . . . . . . . . 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 3434	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
22		eleq1w 2820	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑤 ∈ (𝐹‘𝑥)))
23	22	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥))))
24	23	exbidv 1923	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥))))
25	21, 24	elab 3623	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)))
26		eluniab 4865	. . . . . . . . . 10 ⊢ (𝑤 ∈ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
27	20, 25, 26	3bitr4g 314	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↔ 𝑤 ∈ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)}))
28	27	eqrdv 2735	. . . . . . . 8 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)})
29	28	eleq1d 2822	. . . . . . 7 ⊢ (𝜑 → ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin ↔ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
30	29	adantr 480	. . . . . 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 6669	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝒫 𝐵 → 𝐹 Fn 𝐴)
33		fnrnfv 6900	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)})
36		0ex 5243	. . . . . . . . . . 11 ⊢ ∅ ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ∅ ∈ V)
38		fpwrelmapffslem.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ V
39		fex 7181	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝒫 𝐵 ∧ 𝐴 ∈ V) → 𝐹 ∈ V)
40	31, 38, 39	sylancl 587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
41	40	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → 𝐹 ∈ V)
42	31	ffund 6673	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
43	42	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → Fun 𝐹)
44		opabdm 32684	. . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝒫 𝐵 → 𝐹 ∈ V)
47		suppimacnv 8124	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
48	36, 47	mpan2 692	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ V → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
49	31, 46, 48	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
50	31	feqmptd 6909	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
51	50	cnveqd 5831	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
52	51	imaeq1d 6025	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {∅})) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})))
53	49, 52	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp ∅) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})))
54		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
55	54	mptpreima 6203	. . . . . . . . . . . . . . 15 ⊢ (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})}
56	53, 55	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})})
57		suppvalfn 8118	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅})
58	38, 36, 57	mp3an23 1456	. . . . . . . . . . . . . . . 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 4294	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
61	60	rabbii 3395	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)})
64		df-rab 3391	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
65		19.42v 1955	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)))
66	65	abbii 2804	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
67	64, 66	eqtr4i 2763	. . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
70	45, 69	eqtr4d 2775	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑅 = (𝐹 supp ∅))
71	70	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝑅 ∈ Fin ↔ (𝐹 supp ∅) ∈ Fin))
72	71	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (𝐹 supp ∅) ∈ Fin)
73	37, 41, 43, 72	ffsrn 32801	. . . . . . . . 9 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 ∈ Fin)
74	35, 73	eqeltrrd 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin)
75		unifi 9254	. . . . . . . . 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 412	. . . . . . . 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 9272	. . . . . . 7 ⊢ (∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin)
79	77, 78	impbid1 225	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin ↔ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
80	30, 79	bitr4d 282	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin))
81		opabrn 32685	. . . . . . . 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 2822	. . . . . 6 ⊢ (𝜑 → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin))
84	83	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin))
85	35	sseq1d 3954	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝐹 ⊆ Fin ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin))
86	80, 84, 85	3bitr4d 311	. . . 4 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ ran 𝐹 ⊆ Fin))
87	86	pm5.32da 579	. . 3 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ (dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
88	71	anbi1d 632	. . 3 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
89	87, 88	bitrd 279	. 2 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
90		ancom 460	. . 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 305	1 ⊢ (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∃wrex 3062  {crab 3390  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851  {copab 5148   ↦ cmpt 5167  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634  Rel wrel 5636  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7367   supp csupp 8110  Fincfn 8893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-fin 8897  df-card 9863  df-acn 9866  df-ac 10038

This theorem is referenced by:  fpwrelmapffs  32807