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Theorem fpwrelmapffslem 30644
Description: Lemma for fpwrelmapffs 30646. 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 relopabv 5665 . . . 4 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
3 releq 5622 . . . 4 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}))
42, 3mpbiri 261 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → Rel 𝑅)
5 relfi 30517 . . 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 3163 . . . . . . . . . . . . 13 (∃𝑥𝐴𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑧𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)))
8 ancom 464 . . . . . . . . . . . . . . . 16 ((𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ (𝑤𝑧𝑧 = (𝐹𝑥)))
98exbii 1854 . . . . . . . . . . . . . . 15 (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ ∃𝑧(𝑤𝑧𝑧 = (𝐹𝑥)))
10 fvex 6689 . . . . . . . . . . . . . . . 16 (𝐹𝑥) ∈ V
11 eleq2 2821 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑥) → (𝑤𝑧𝑤 ∈ (𝐹𝑥)))
1210, 11ceqsexv 3445 . . . . . . . . . . . . . . 15 (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ 𝑤 ∈ (𝐹𝑥))
139, 12bitr3i 280 . . . . . . . . . . . . . 14 (∃𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ 𝑤 ∈ (𝐹𝑥))
1413rexbii 3161 . . . . . . . . . . . . 13 (∃𝑥𝐴𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑥𝐴 𝑤 ∈ (𝐹𝑥))
15 r19.42v 3254 . . . . . . . . . . . . . 14 (∃𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)) ↔ (𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
1615exbii 1854 . . . . . . . . . . . . 13 (∃𝑧𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
177, 14, 163bitr3ri 305 . . . . . . . . . . . 12 (∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)) ↔ ∃𝑥𝐴 𝑤 ∈ (𝐹𝑥))
18 df-rex 3059 . . . . . . . . . . . 12 (∃𝑥𝐴 𝑤 ∈ (𝐹𝑥) ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)))
1917, 18bitr2i 279 . . . . . . . . . . 11 (∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
2019a1i 11 . . . . . . . . . 10 (𝜑 → (∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥))))
21 vex 3402 . . . . . . . . . . 11 𝑤 ∈ V
22 eleq1w 2815 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑦 ∈ (𝐹𝑥) ↔ 𝑤 ∈ (𝐹𝑥)))
2322anbi2d 632 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ (𝑥𝐴𝑤 ∈ (𝐹𝑥))))
2423exbidv 1928 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥))))
2521, 24elab 3573 . . . . . . . . . 10 (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)))
26 eluniab 4811 . . . . . . . . . 10 (𝑤 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
2720, 25, 263bitr4g 317 . . . . . . . . 9 (𝜑 → (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ 𝑤 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)}))
2827eqrdv 2736 . . . . . . . 8 (𝜑 → {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
2928eleq1d 2817 . . . . . . 7 (𝜑 → ({𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
3029adantr 484 . . . . . 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 6504 . . . . . . . . . . 11 (𝐹:𝐴⟶𝒫 𝐵𝐹 Fn 𝐴)
33 fnrnfv 6731 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
3431, 32, 333syl 18 . . . . . . . . . 10 (𝜑 → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
3534adantr 484 . . . . . . . . 9 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
36 0ex 5175 . . . . . . . . . . 11 ∅ ∈ V
3736a1i 11 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ∅ ∈ V)
38 fpwrelmapffslem.1 . . . . . . . . . . . 12 𝐴 ∈ V
39 fex 7001 . . . . . . . . . . . 12 ((𝐹:𝐴⟶𝒫 𝐵𝐴 ∈ V) → 𝐹 ∈ V)
4031, 38, 39sylancl 589 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
4140adantr 484 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → 𝐹 ∈ V)
4231ffund 6508 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
4342adantr 484 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → Fun 𝐹)
44 opabdm 30527 . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . 17 (𝐹:𝐴⟶𝒫 𝐵𝐹 ∈ V)
47 suppimacnv 7871 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
4836, 47mpan2 691 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ V → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
4931, 46, 483syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
5031feqmptd 6739 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
5150cnveqd 5718 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
5251imaeq1d 5902 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 “ (V ∖ {∅})) = ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})))
5349, 52eqtrd 2773 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 supp ∅) = ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})))
54 eqid 2738 . . . . . . . . . . . . . . . 16 (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐹𝑥))
5554mptpreima 6070 . . . . . . . . . . . . . . 15 ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})}
5653, 55eqtrdi 2789 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})})
57 suppvalfn 7866 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴𝐴 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
5838, 36, 57mp3an23 1454 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
5931, 32, 583syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
60 n0 4235 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑥))
6160rabbii 3374 . . . . . . . . . . . . . . . 16 {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)}
6261a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)})
6359, 56, 623eqtr3d 2781 . . . . . . . . . . . . . 14 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)})
64 df-rab 3062 . . . . . . . . . . . . . . . 16 {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
65 19.42v 1961 . . . . . . . . . . . . . . . . 17 (∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥)))
6665abbii 2803 . . . . . . . . . . . . . . . 16 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
6764, 66eqtr4i 2764 . . . . . . . . . . . . . . 15 {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))}
6867a1i 11 . . . . . . . . . . . . . 14 (𝜑 → {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
6956, 63, 683eqtrd 2777 . . . . . . . . . . . . 13 (𝜑 → (𝐹 supp ∅) = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
7045, 69eqtr4d 2776 . . . . . . . . . . . 12 (𝜑 → dom 𝑅 = (𝐹 supp ∅))
7170eleq1d 2817 . . . . . . . . . . 11 (𝜑 → (dom 𝑅 ∈ Fin ↔ (𝐹 supp ∅) ∈ Fin))
7271biimpa 480 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (𝐹 supp ∅) ∈ Fin)
7337, 41, 43, 72ffsrn 30641 . . . . . . . . 9 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 ∈ Fin)
7435, 73eqeltrrd 2834 . . . . . . . 8 ((𝜑 ∧ dom 𝑅 ∈ Fin) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin)
75 unifi 8888 . . . . . . . . 9 (({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin ∧ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin)
7675ex 416 . . . . . . . 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 30624 . . . . . . 7 ( {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin)
7977, 78impbid1 228 . . . . . 6 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
8030, 79bitr4d 285 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin))
81 opabrn 30528 . . . . . . . 8 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
821, 81syl 17 . . . . . . 7 (𝜑 → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
8382eleq1d 2817 . . . . . 6 (𝜑 → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin))
8483adantr 484 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin))
8535sseq1d 3908 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝐹 ⊆ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin))
8680, 84, 853bitr4d 314 . . . 4 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ ran 𝐹 ⊆ Fin))
8786pm5.32da 582 . . 3 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ (dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
8871anbi1d 633 . . 3 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
8987, 88bitrd 282 . 2 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
90 ancom 464 . . 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 308 1 (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2114  {cab 2716  wne 2934  wrex 3054  {crab 3057  Vcvv 3398  cdif 3840  wss 3843  c0 4211  𝒫 cpw 4488  {csn 4516   cuni 4796  {copab 5092  cmpt 5110  ccnv 5524  dom cdm 5525  ran crn 5526  cima 5528  Rel wrel 5530  Fun wfun 6333   Fn wfn 6334  wf 6335  cfv 6339  (class class class)co 7172   supp csupp 7858  Fincfn 8557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-ac2 9965
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-om 7602  df-1st 7716  df-2nd 7717  df-supp 7859  df-wrecs 7978  df-recs 8039  df-1o 8133  df-er 8322  df-map 8441  df-en 8558  df-dom 8559  df-fin 8561  df-card 9443  df-acn 9446  df-ac 9618
This theorem is referenced by:  fpwrelmapffs  30646
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