Theorem fpwrelmapffslem 28895

Description: Lemma for fpwrelmapffs 28897. 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		relopab 5169	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
3		releq 5124	. . . 4 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}))
4	2, 3	mpbiri 247	. . 3 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → Rel 𝑅)
5		relfi 28797	. . 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 3198	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
8		ancom 465	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ↔ (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
9	8	exbii 1764	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
10		fvex 6113	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑥) ∈ V
11		eleq2 2677	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ (𝐹‘𝑥)))
12	10, 11	ceqsexv 3215	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ↔ 𝑤 ∈ (𝐹‘𝑥))
13	9, 12	bitr3i 265	. . . . . . . . . . . . . 14 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ 𝑤 ∈ (𝐹‘𝑥))
14	13	rexbii 3023	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥))
15		r19.42v 3073	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
16	15	exbii 1764	. . . . . . . . . . . . 13 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
17	7, 14, 16	3bitr3ri 290	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥))
18		df-rex 2902	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)))
19	17, 18	bitr2i 264	. . . . . . . . . . 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 3176	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
22		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑤 ∈ (𝐹‘𝑥)))
23	22	anbi2d 736	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥))))
24	23	exbidv 1837	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥))))
25	21, 24	elab 3319	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)))
26		eluniab 4383	. . . . . . . . . 10 ⊢ (𝑤 ∈ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)))
27	20, 25, 26	3bitr4g 302	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↔ 𝑤 ∈ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)}))
28	27	eqrdv 2608	. . . . . . . 8 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)})
29	28	eleq1d 2672	. . . . . . 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 5958	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝒫 𝐵 → 𝐹 Fn 𝐴)
33		fnrnfv 6152	. . . . . . . . . . 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 4718	. . . . . . . . . . 11 ⊢ ∅ ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ∅ ∈ V)
38		fpwrelmapffslem.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ V
39		fex 6394	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝒫 𝐵 ∧ 𝐴 ∈ V) → 𝐹 ∈ V)
40	31, 38, 39	sylancl 693	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
41	40	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → 𝐹 ∈ V)
42		ffun 5961	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝒫 𝐵 → Fun 𝐹)
43	31, 42	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → Fun 𝐹)
45		opabdm 28803	. . . . . . . . . . . . . 14 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → dom 𝑅 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
46	1, 45	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝑅 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
47	38, 39	mpan2 703	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝒫 𝐵 → 𝐹 ∈ V)
48		suppimacnv 7193	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
49	36, 48	mpan2 703	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ V → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
50	31, 47, 49	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
51	31	feqmptd 6159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
52	51	cnveqd 5220	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
53	52	imaeq1d 5384	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {∅})) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})))
54	50, 53	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp ∅) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})))
55		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
56	55	mptpreima 5545	. . . . . . . . . . . . . . 15 ⊢ (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})}
57	54, 56	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})})
58		suppvalfn 7189	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅})
59	38, 36, 58	mp3an23 1408	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn 𝐴 → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅})
60	31, 32, 59	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅})
61		n0 3890	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
62	61	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)))
63	62	rabbiia 3161	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)}
64	63	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)})
65	60, 57, 64	3eqtr3d 2652	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)})
66		df-rab 2905	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
67		19.42v 1905	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)))
68	67	abbii 2726	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
69	66, 68	eqtr4i 2635	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
70	69	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
71	57, 65, 70	3eqtrd 2648	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
72	46, 71	eqtr4d 2647	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑅 = (𝐹 supp ∅))
73	72	eleq1d 2672	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝑅 ∈ Fin ↔ (𝐹 supp ∅) ∈ Fin))
74	73	biimpa 500	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (𝐹 supp ∅) ∈ Fin)
75	37, 41, 44, 74	ffsrn 28892	. . . . . . . . 9 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 ∈ Fin)
76	35, 75	eqeltrrd 2689	. . . . . . . 8 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin)
77		unifi 8138	. . . . . . . . 9 ⊢ (({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin)
78	77	ex 449	. . . . . . . 8 ⊢ ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
79	76, 78	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
80		unifi3 28873	. . . . . . 7 ⊢ (∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin)
81	79, 80	impbid1 214	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin ↔ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
82	30, 81	bitr4d 270	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin))
83		opabrn 28804	. . . . . . . 8 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
84	1, 83	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
85	84	eleq1d 2672	. . . . . 6 ⊢ (𝜑 → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin))
86	85	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin))
87	35	sseq1d 3595	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝐹 ⊆ Fin ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin))
88	82, 86, 87	3bitr4d 299	. . . 4 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ ran 𝐹 ⊆ Fin))
89	88	pm5.32da 671	. . 3 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ (dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
90	73	anbi1d 737	. . 3 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
91	89, 90	bitrd 267	. 2 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
92		ancom 465	. . 3 ⊢ (((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin))
93	92	a1i 11	. 2 ⊢ (𝜑 → (((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
94	6, 91, 93	3bitrd 293	1 ⊢ (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  {copab 4642   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   supp csupp 7182  Fincfn 7841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-ac2 9168

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-fin 7845  df-card 8648  df-acn 8651  df-ac 8822

This theorem is referenced by:  fpwrelmapffs  28897