Theorem fpwrelmapffslem 31067

Description: Lemma for fpwrelmapffs 31069. 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 5731	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
3		releq 5687	. . . 4 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}))
4	2, 3	mpbiri 257	. . 3 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} → Rel 𝑅)
5		relfi 30941	. . 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 3233	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)))
8		ancom 461	. . . . . . . . . . . . . . . 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 6787	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑥) ∈ V
11		eleq2 2827	. . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . 14 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ 𝑤 ∈ (𝐹‘𝑥))
14	13	rexbii 3181	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥))
15		r19.42v 3279	. . . . . . . . . . . . . 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 3070	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ (𝐹‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)))
19	17, 18	bitr2i 275	. . . . . . . . . . 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 3436	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
22		eleq1w 2821	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑤 ∈ (𝐹‘𝑥)))
23	22	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥))))
24	23	exbidv 1924	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥))))
25	21, 24	elab 3609	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ (𝐹‘𝑥)))
26		eluniab 4854	. . . . . . . . . 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 2736	. . . . . . . 8 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)})
29	28	eleq1d 2823	. . . . . . 7 ⊢ (𝜑 → ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin ↔ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
30	29	adantr 481	. . . . . 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 6600	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝒫 𝐵 → 𝐹 Fn 𝐴)
33		fnrnfv 6829	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)})
36		0ex 5231	. . . . . . . . . . 11 ⊢ ∅ ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ∅ ∈ V)
38		fpwrelmapffslem.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ V
39		fex 7102	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝒫 𝐵 ∧ 𝐴 ∈ V) → 𝐹 ∈ V)
40	31, 38, 39	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
41	40	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → 𝐹 ∈ V)
42	31	ffund 6604	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
43	42	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → Fun 𝐹)
44		opabdm 30951	. . . . . . . . . . . . . 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 688	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝒫 𝐵 → 𝐹 ∈ V)
47		suppimacnv 7990	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
48	36, 47	mpan2 688	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ V → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
49	31, 46, 48	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 supp ∅) = (◡𝐹 “ (V ∖ {∅})))
50	31	feqmptd 6837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
51	50	cnveqd 5784	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
52	51	imaeq1d 5968	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {∅})) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})))
53	49, 52	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp ∅) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})))
54		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
55	54	mptpreima 6141	. . . . . . . . . . . . . . 15 ⊢ (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ (V ∖ {∅})) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})}
56	53, 55	eqtrdi 2794	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})})
57		suppvalfn 7985	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ ∅})
58	38, 36, 57	mp3an23 1452	. . . . . . . . . . . . . . . 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 4280	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
61	60	rabbii 3408	. . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (V ∖ {∅})} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)})
64		df-rab 3073	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
65		19.42v 1957	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)))
66	65	abbii 2808	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
67	64, 66	eqtr4i 2769	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp ∅) = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})
70	45, 69	eqtr4d 2781	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑅 = (𝐹 supp ∅))
71	70	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝑅 ∈ Fin ↔ (𝐹 supp ∅) ∈ Fin))
72	71	biimpa 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (𝐹 supp ∅) ∈ Fin)
73	37, 41, 43, 72	ffsrn 31064	. . . . . . . . 9 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 ∈ Fin)
74	35, 73	eqeltrrd 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin)
75		unifi 9108	. . . . . . . . 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 413	. . . . . . . 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 31047	. . . . . . 7 ⊢ (∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin)
79	77, 78	impbid1 224	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin ↔ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ∈ Fin))
80	30, 79	bitr4d 281	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝐹‘𝑥)} ⊆ Fin))
81		opabrn 30952	. . . . . . . 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 2823	. . . . . 6 ⊢ (𝜑 → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin))
84	83	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ∈ Fin))
85	35	sseq1d 3952	. . . . 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 630	. . 3 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
89	87, 88	bitrd 278	. 2 ⊢ (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
90		ancom 461	. . 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 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∃wrex 3065  {crab 3068  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839  {copab 5136   ↦ cmpt 5157  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592  Rel wrel 5594  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   supp csupp 7977  Fincfn 8733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-ac2 10219

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-fin 8737  df-card 9697  df-acn 9700  df-ac 9872

This theorem is referenced by:  fpwrelmapffs  31069