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Theorem fpwrelmapffslem 29891
Description: Lemma for fpwrelmapffs 29893. 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 5416 . . . 4 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
3 releq 5371 . . . 4 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}))
42, 3mpbiri 249 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → Rel 𝑅)
5 relfi 29798 . . 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 3378 . . . . . . . . . . . . 13 (∃𝑥𝐴𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑧𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)))
8 ancom 452 . . . . . . . . . . . . . . . 16 ((𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ (𝑤𝑧𝑧 = (𝐹𝑥)))
98exbii 1943 . . . . . . . . . . . . . . 15 (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ ∃𝑧(𝑤𝑧𝑧 = (𝐹𝑥)))
10 fvex 6388 . . . . . . . . . . . . . . . 16 (𝐹𝑥) ∈ V
11 eleq2 2833 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑥) → (𝑤𝑧𝑤 ∈ (𝐹𝑥)))
1210, 11ceqsexv 3395 . . . . . . . . . . . . . . 15 (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ 𝑤 ∈ (𝐹𝑥))
139, 12bitr3i 268 . . . . . . . . . . . . . 14 (∃𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ 𝑤 ∈ (𝐹𝑥))
1413rexbii 3188 . . . . . . . . . . . . 13 (∃𝑥𝐴𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑥𝐴 𝑤 ∈ (𝐹𝑥))
15 r19.42v 3239 . . . . . . . . . . . . . 14 (∃𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)) ↔ (𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
1615exbii 1943 . . . . . . . . . . . . 13 (∃𝑧𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
177, 14, 163bitr3ri 293 . . . . . . . . . . . 12 (∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)) ↔ ∃𝑥𝐴 𝑤 ∈ (𝐹𝑥))
18 df-rex 3061 . . . . . . . . . . . 12 (∃𝑥𝐴 𝑤 ∈ (𝐹𝑥) ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)))
1917, 18bitr2i 267 . . . . . . . . . . 11 (∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
2019a1i 11 . . . . . . . . . 10 (𝜑 → (∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥))))
21 vex 3353 . . . . . . . . . . 11 𝑤 ∈ V
22 eleq1w 2827 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑦 ∈ (𝐹𝑥) ↔ 𝑤 ∈ (𝐹𝑥)))
2322anbi2d 622 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ (𝑥𝐴𝑤 ∈ (𝐹𝑥))))
2423exbidv 2016 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥))))
2521, 24elab 3504 . . . . . . . . . 10 (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)))
26 eluniab 4605 . . . . . . . . . 10 (𝑤 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
2720, 25, 263bitr4g 305 . . . . . . . . 9 (𝜑 → (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ 𝑤 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)}))
2827eqrdv 2763 . . . . . . . 8 (𝜑 → {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
2928eleq1d 2829 . . . . . . 7 (𝜑 → ({𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
3029adantr 472 . . . . . 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 6223 . . . . . . . . . . 11 (𝐹:𝐴⟶𝒫 𝐵𝐹 Fn 𝐴)
33 fnrnfv 6431 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
3431, 32, 333syl 18 . . . . . . . . . 10 (𝜑 → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
3534adantr 472 . . . . . . . . 9 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
36 0ex 4950 . . . . . . . . . . 11 ∅ ∈ V
3736a1i 11 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ∅ ∈ V)
38 fpwrelmapffslem.1 . . . . . . . . . . . 12 𝐴 ∈ V
39 fex 6682 . . . . . . . . . . . 12 ((𝐹:𝐴⟶𝒫 𝐵𝐴 ∈ V) → 𝐹 ∈ V)
4031, 38, 39sylancl 580 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
4140adantr 472 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → 𝐹 ∈ V)
4231ffund 6227 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
4342adantr 472 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → Fun 𝐹)
44 opabdm 29806 . . . . . . . . . . . . . 14 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → dom 𝑅 = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
451, 44syl 17 . . . . . . . . . . . . 13 (𝜑 → dom 𝑅 = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
4638, 39mpan2 682 . . . . . . . . . . . . . . . . 17 (𝐹:𝐴⟶𝒫 𝐵𝐹 ∈ V)
47 suppimacnv 7508 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
4836, 47mpan2 682 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ V → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
4931, 46, 483syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
5031feqmptd 6438 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
5150cnveqd 5466 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
5251imaeq1d 5647 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 “ (V ∖ {∅})) = ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})))
5349, 52eqtrd 2799 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 supp ∅) = ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})))
54 eqid 2765 . . . . . . . . . . . . . . . 16 (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐹𝑥))
5554mptpreima 5814 . . . . . . . . . . . . . . 15 ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})}
5653, 55syl6eq 2815 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})})
57 suppvalfn 7504 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴𝐴 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
5838, 36, 57mp3an23 1577 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
5931, 32, 583syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
60 n0 4095 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑥))
6160rabbii 3334 . . . . . . . . . . . . . . . 16 {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)}
6261a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)})
6359, 56, 623eqtr3d 2807 . . . . . . . . . . . . . 14 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)})
64 df-rab 3064 . . . . . . . . . . . . . . . 16 {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
65 19.42v 2047 . . . . . . . . . . . . . . . . 17 (∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥)))
6665abbii 2882 . . . . . . . . . . . . . . . 16 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
6764, 66eqtr4i 2790 . . . . . . . . . . . . . . 15 {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))}
6867a1i 11 . . . . . . . . . . . . . 14 (𝜑 → {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
6956, 63, 683eqtrd 2803 . . . . . . . . . . . . 13 (𝜑 → (𝐹 supp ∅) = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
7045, 69eqtr4d 2802 . . . . . . . . . . . 12 (𝜑 → dom 𝑅 = (𝐹 supp ∅))
7170eleq1d 2829 . . . . . . . . . . 11 (𝜑 → (dom 𝑅 ∈ Fin ↔ (𝐹 supp ∅) ∈ Fin))
7271biimpa 468 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (𝐹 supp ∅) ∈ Fin)
7337, 41, 43, 72ffsrn 29888 . . . . . . . . 9 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 ∈ Fin)
7435, 73eqeltrrd 2845 . . . . . . . 8 ((𝜑 ∧ dom 𝑅 ∈ Fin) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin)
75 unifi 8462 . . . . . . . . 9 (({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin ∧ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin)
7675ex 401 . . . . . . . 8 ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin → ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
7774, 76syl 17 . . . . . . 7 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
78 unifi3 29874 . . . . . . 7 ( {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin)
7977, 78impbid1 216 . . . . . 6 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
8030, 79bitr4d 273 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin))
81 opabrn 29807 . . . . . . . 8 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
821, 81syl 17 . . . . . . 7 (𝜑 → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
8382eleq1d 2829 . . . . . 6 (𝜑 → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin))
8483adantr 472 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin))
8535sseq1d 3792 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝐹 ⊆ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin))
8680, 84, 853bitr4d 302 . . . 4 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ ran 𝐹 ⊆ Fin))
8786pm5.32da 574 . . 3 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ (dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
8871anbi1d 623 . . 3 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
8987, 88bitrd 270 . 2 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
90 ancom 452 . . 3 (((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin))
9190a1i 11 . 2 (𝜑 → (((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
926, 89, 913bitrd 296 1 (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wrex 3056  {crab 3059  Vcvv 3350  cdif 3729  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334   cuni 4594  {copab 4871  cmpt 4888  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280  Rel wrel 5282  Fun wfun 6062   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842   supp csupp 7497  Fincfn 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-ac2 9538
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-fin 8164  df-card 9016  df-acn 9019  df-ac 9190
This theorem is referenced by:  fpwrelmapffs  29893
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