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Theorem fpwrelmapffslem 29341
Description: Lemma for fpwrelmapffs 29343. 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 (𝜑𝐹:𝐴⟶𝒫 𝐵)
fpwrelmapffslem.4 (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})
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.
StepHypRef Expression
1 fpwrelmapffslem.4 . . 3 (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})
2 relopab 5212 . . . 4 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
3 releq 5167 . . . 4 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}))
42, 3mpbiri 248 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → Rel 𝑅)
5 relfi 29251 . . 3 (Rel 𝑅 → (𝑅 ∈ Fin ↔ (dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin)))
61, 4, 53syl 18 . 2 (𝜑 → (𝑅 ∈ Fin ↔ (dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin)))
7 rexcom4 3216 . . . . . . . . . . . . 13 (∃𝑥𝐴𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑧𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)))
8 ancom 466 . . . . . . . . . . . . . . . 16 ((𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ (𝑤𝑧𝑧 = (𝐹𝑥)))
98exbii 1772 . . . . . . . . . . . . . . 15 (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ ∃𝑧(𝑤𝑧𝑧 = (𝐹𝑥)))
10 fvex 6160 . . . . . . . . . . . . . . . 16 (𝐹𝑥) ∈ V
11 eleq2 2693 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑥) → (𝑤𝑧𝑤 ∈ (𝐹𝑥)))
1210, 11ceqsexv 3233 . . . . . . . . . . . . . . 15 (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ 𝑤 ∈ (𝐹𝑥))
139, 12bitr3i 266 . . . . . . . . . . . . . 14 (∃𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ 𝑤 ∈ (𝐹𝑥))
1413rexbii 3039 . . . . . . . . . . . . 13 (∃𝑥𝐴𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑥𝐴 𝑤 ∈ (𝐹𝑥))
15 r19.42v 3089 . . . . . . . . . . . . . 14 (∃𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)) ↔ (𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
1615exbii 1772 . . . . . . . . . . . . 13 (∃𝑧𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
177, 14, 163bitr3ri 291 . . . . . . . . . . . 12 (∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)) ↔ ∃𝑥𝐴 𝑤 ∈ (𝐹𝑥))
18 df-rex 2918 . . . . . . . . . . . 12 (∃𝑥𝐴 𝑤 ∈ (𝐹𝑥) ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)))
1917, 18bitr2i 265 . . . . . . . . . . 11 (∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
2019a1i 11 . . . . . . . . . 10 (𝜑 → (∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥))))
21 vex 3194 . . . . . . . . . . 11 𝑤 ∈ V
22 eleq1 2692 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑦 ∈ (𝐹𝑥) ↔ 𝑤 ∈ (𝐹𝑥)))
2322anbi2d 739 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ (𝑥𝐴𝑤 ∈ (𝐹𝑥))))
2423exbidv 1852 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥))))
2521, 24elab 3338 . . . . . . . . . 10 (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)))
26 eluniab 4418 . . . . . . . . . 10 (𝑤 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
2720, 25, 263bitr4g 303 . . . . . . . . 9 (𝜑 → (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ 𝑤 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)}))
2827eqrdv 2624 . . . . . . . 8 (𝜑 → {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
2928eleq1d 2688 . . . . . . 7 (𝜑 → ({𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
3029adantr 481 . . . . . 6 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
31 fpwrelmapffslem.3 . . . . . . . . . . 11 (𝜑𝐹:𝐴⟶𝒫 𝐵)
32 ffn 6004 . . . . . . . . . . 11 (𝐹:𝐴⟶𝒫 𝐵𝐹 Fn 𝐴)
33 fnrnfv 6200 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
3431, 32, 333syl 18 . . . . . . . . . 10 (𝜑 → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
3534adantr 481 . . . . . . . . 9 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
36 0ex 4755 . . . . . . . . . . 11 ∅ ∈ V
3736a1i 11 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ∅ ∈ V)
38 fpwrelmapffslem.1 . . . . . . . . . . . 12 𝐴 ∈ V
39 fex 6445 . . . . . . . . . . . 12 ((𝐹:𝐴⟶𝒫 𝐵𝐴 ∈ V) → 𝐹 ∈ V)
4031, 38, 39sylancl 693 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
4140adantr 481 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → 𝐹 ∈ V)
42 ffun 6007 . . . . . . . . . . . 12 (𝐹:𝐴⟶𝒫 𝐵 → Fun 𝐹)
4331, 42syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
4443adantr 481 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → Fun 𝐹)
45 opabdm 29257 . . . . . . . . . . . . . 14 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → dom 𝑅 = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
461, 45syl 17 . . . . . . . . . . . . 13 (𝜑 → dom 𝑅 = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
4738, 39mpan2 706 . . . . . . . . . . . . . . . . 17 (𝐹:𝐴⟶𝒫 𝐵𝐹 ∈ V)
48 suppimacnv 7252 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
4936, 48mpan2 706 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ V → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
5031, 47, 493syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
5131feqmptd 6207 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
5251cnveqd 5263 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
5352imaeq1d 5428 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 “ (V ∖ {∅})) = ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})))
5450, 53eqtrd 2660 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 supp ∅) = ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})))
55 eqid 2626 . . . . . . . . . . . . . . . 16 (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐹𝑥))
5655mptpreima 5590 . . . . . . . . . . . . . . 15 ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})}
5754, 56syl6eq 2676 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})})
58 suppvalfn 7248 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴𝐴 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
5938, 36, 58mp3an23 1413 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
6031, 32, 593syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
61 n0 3912 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑥))
6261a1i 11 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝐹𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑥)))
6362rabbiia 3178 . . . . . . . . . . . . . . . 16 {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)}
6463a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)})
6560, 57, 643eqtr3d 2668 . . . . . . . . . . . . . 14 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)})
66 df-rab 2921 . . . . . . . . . . . . . . . 16 {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
67 19.42v 1920 . . . . . . . . . . . . . . . . 17 (∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥)))
6867abbii 2742 . . . . . . . . . . . . . . . 16 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
6966, 68eqtr4i 2651 . . . . . . . . . . . . . . 15 {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))}
7069a1i 11 . . . . . . . . . . . . . 14 (𝜑 → {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
7157, 65, 703eqtrd 2664 . . . . . . . . . . . . 13 (𝜑 → (𝐹 supp ∅) = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
7246, 71eqtr4d 2663 . . . . . . . . . . . 12 (𝜑 → dom 𝑅 = (𝐹 supp ∅))
7372eleq1d 2688 . . . . . . . . . . 11 (𝜑 → (dom 𝑅 ∈ Fin ↔ (𝐹 supp ∅) ∈ Fin))
7473biimpa 501 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (𝐹 supp ∅) ∈ Fin)
7537, 41, 44, 74ffsrn 29338 . . . . . . . . 9 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 ∈ Fin)
7635, 75eqeltrrd 2705 . . . . . . . 8 ((𝜑 ∧ dom 𝑅 ∈ Fin) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin)
77 unifi 8200 . . . . . . . . 9 (({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin ∧ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin)
7877ex 450 . . . . . . . 8 ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin → ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
7976, 78syl 17 . . . . . . 7 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
80 unifi3 29324 . . . . . . 7 ( {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin)
8179, 80impbid1 215 . . . . . 6 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
8230, 81bitr4d 271 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin))
83 opabrn 29258 . . . . . . . 8 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
841, 83syl 17 . . . . . . 7 (𝜑 → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
8584eleq1d 2688 . . . . . 6 (𝜑 → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin))
8685adantr 481 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin))
8735sseq1d 3616 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝐹 ⊆ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin))
8882, 86, 873bitr4d 300 . . . 4 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ ran 𝐹 ⊆ Fin))
8988pm5.32da 672 . . 3 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ (dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
9073anbi1d 740 . . 3 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
9189, 90bitrd 268 . 2 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
92 ancom 466 . . 3 (((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin))
9392a1i 11 . 2 (𝜑 → (((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
946, 91, 933bitrd 294 1 (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  {cab 2612  wne 2796  wrex 2913  {crab 2916  Vcvv 3191  cdif 3557  wss 3560  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407  {copab 4677  cmpt 4678  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  Rel wrel 5084  Fun wfun 5844   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605   supp csupp 7241  Fincfn 7900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-ac2 9230
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-fin 7904  df-card 8710  df-acn 8713  df-ac 8884
This theorem is referenced by:  fpwrelmapffs  29343
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