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Theorem fpwrelmapffslem 30381
Description: Lemma for fpwrelmapffs 30383. 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 5695 . . . 4 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
3 releq 5650 . . . 4 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}))
42, 3mpbiri 259 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → Rel 𝑅)
5 relfi 30267 . . 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 3254 . . . . . . . . . . . . 13 (∃𝑥𝐴𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑧𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)))
8 ancom 461 . . . . . . . . . . . . . . . 16 ((𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ (𝑤𝑧𝑧 = (𝐹𝑥)))
98exbii 1841 . . . . . . . . . . . . . . 15 (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ ∃𝑧(𝑤𝑧𝑧 = (𝐹𝑥)))
10 fvex 6680 . . . . . . . . . . . . . . . 16 (𝐹𝑥) ∈ V
11 eleq2 2906 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑥) → (𝑤𝑧𝑤 ∈ (𝐹𝑥)))
1210, 11ceqsexv 3547 . . . . . . . . . . . . . . 15 (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑤𝑧) ↔ 𝑤 ∈ (𝐹𝑥))
139, 12bitr3i 278 . . . . . . . . . . . . . 14 (∃𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ 𝑤 ∈ (𝐹𝑥))
1413rexbii 3252 . . . . . . . . . . . . 13 (∃𝑥𝐴𝑧(𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑥𝐴 𝑤 ∈ (𝐹𝑥))
15 r19.42v 3355 . . . . . . . . . . . . . 14 (∃𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)) ↔ (𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
1615exbii 1841 . . . . . . . . . . . . 13 (∃𝑧𝑥𝐴 (𝑤𝑧𝑧 = (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
177, 14, 163bitr3ri 303 . . . . . . . . . . . 12 (∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)) ↔ ∃𝑥𝐴 𝑤 ∈ (𝐹𝑥))
18 df-rex 3149 . . . . . . . . . . . 12 (∃𝑥𝐴 𝑤 ∈ (𝐹𝑥) ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)))
1917, 18bitr2i 277 . . . . . . . . . . 11 (∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
2019a1i 11 . . . . . . . . . 10 (𝜑 → (∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)) ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥))))
21 vex 3503 . . . . . . . . . . 11 𝑤 ∈ V
22 eleq1w 2900 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑦 ∈ (𝐹𝑥) ↔ 𝑤 ∈ (𝐹𝑥)))
2322anbi2d 628 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ (𝑥𝐴𝑤 ∈ (𝐹𝑥))))
2423exbidv 1915 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥))))
2521, 24elab 3671 . . . . . . . . . 10 (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ ∃𝑥(𝑥𝐴𝑤 ∈ (𝐹𝑥)))
26 eluniab 4848 . . . . . . . . . 10 (𝑤 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ↔ ∃𝑧(𝑤𝑧 ∧ ∃𝑥𝐴 𝑧 = (𝐹𝑥)))
2720, 25, 263bitr4g 315 . . . . . . . . 9 (𝜑 → (𝑤 ∈ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ 𝑤 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)}))
2827eqrdv 2824 . . . . . . . 8 (𝜑 → {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)})
2928eleq1d 2902 . . . . . . 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 6511 . . . . . . . . . . 11 (𝐹:𝐴⟶𝒫 𝐵𝐹 Fn 𝐴)
33 fnrnfv 6722 . . . . . . . . . . 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 5208 . . . . . . . . . . 11 ∅ ∈ V
3736a1i 11 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ∅ ∈ V)
38 fpwrelmapffslem.1 . . . . . . . . . . . 12 𝐴 ∈ V
39 fex 6984 . . . . . . . . . . . 12 ((𝐹:𝐴⟶𝒫 𝐵𝐴 ∈ V) → 𝐹 ∈ V)
4031, 38, 39sylancl 586 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
4140adantr 481 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → 𝐹 ∈ V)
4231ffund 6515 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
4342adantr 481 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → Fun 𝐹)
44 opabdm 30277 . . . . . . . . . . . . . 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 687 . . . . . . . . . . . . . . . . 17 (𝐹:𝐴⟶𝒫 𝐵𝐹 ∈ V)
47 suppimacnv 7832 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
4836, 47mpan2 687 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ V → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
4931, 46, 483syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 supp ∅) = (𝐹 “ (V ∖ {∅})))
5031feqmptd 6730 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
5150cnveqd 5745 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
5251imaeq1d 5926 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 “ (V ∖ {∅})) = ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})))
5349, 52eqtrd 2861 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 supp ∅) = ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})))
54 eqid 2826 . . . . . . . . . . . . . . . 16 (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐹𝑥))
5554mptpreima 6090 . . . . . . . . . . . . . . 15 ((𝑥𝐴 ↦ (𝐹𝑥)) “ (V ∖ {∅})) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})}
5653, 55syl6eq 2877 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})})
57 suppvalfn 7828 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴𝐴 ∈ V ∧ ∅ ∈ V) → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
5838, 36, 57mp3an23 1446 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
5931, 32, 583syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 supp ∅) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅})
60 n0 4314 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑥))
6160rabbii 3479 . . . . . . . . . . . . . . . 16 {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)}
6261a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) ≠ ∅} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)})
6359, 56, 623eqtr3d 2869 . . . . . . . . . . . . . 14 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) ∈ (V ∖ {∅})} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)})
64 df-rab 3152 . . . . . . . . . . . . . . . 16 {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
65 19.42v 1947 . . . . . . . . . . . . . . . . 17 (∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥)))
6665abbii 2891 . . . . . . . . . . . . . . . 16 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
6764, 66eqtr4i 2852 . . . . . . . . . . . . . . 15 {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))}
6867a1i 11 . . . . . . . . . . . . . 14 (𝜑 → {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
6956, 63, 683eqtrd 2865 . . . . . . . . . . . . 13 (𝜑 → (𝐹 supp ∅) = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
7045, 69eqtr4d 2864 . . . . . . . . . . . 12 (𝜑 → dom 𝑅 = (𝐹 supp ∅))
7170eleq1d 2902 . . . . . . . . . . 11 (𝜑 → (dom 𝑅 ∈ Fin ↔ (𝐹 supp ∅) ∈ Fin))
7271biimpa 477 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (𝐹 supp ∅) ∈ Fin)
7337, 41, 43, 72ffsrn 30378 . . . . . . . . 9 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ran 𝐹 ∈ Fin)
7435, 73eqeltrrd 2919 . . . . . . . 8 ((𝜑 ∧ dom 𝑅 ∈ Fin) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin)
75 unifi 8802 . . . . . . . . 9 (({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin ∧ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin)
7675ex 413 . . . . . . . 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 30361 . . . . . . 7 ( {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin)
7977, 78impbid1 226 . . . . . 6 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ∈ Fin))
8030, 79bitr4d 283 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → ({𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin))
81 opabrn 30278 . . . . . . . 8 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
821, 81syl 17 . . . . . . 7 (𝜑 → ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))})
8382eleq1d 2902 . . . . . 6 (𝜑 → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin))
8483adantr 481 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ∈ Fin))
8535sseq1d 4002 . . . . 5 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝐹 ⊆ Fin ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹𝑥)} ⊆ Fin))
8680, 84, 853bitr4d 312 . . . 4 ((𝜑 ∧ dom 𝑅 ∈ Fin) → (ran 𝑅 ∈ Fin ↔ ran 𝐹 ⊆ Fin))
8786pm5.32da 579 . . 3 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin) ↔ (dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
8871anbi1d 629 . . 3 (𝜑 → ((dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ ((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin)))
8987, 88bitrd 280 . 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))
9190a1i 11 . 2 (𝜑 → (((𝐹 supp ∅) ∈ Fin ∧ ran 𝐹 ⊆ Fin) ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
926, 89, 913bitrd 306 1 (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  {cab 2804  wne 3021  wrex 3144  {crab 3147  Vcvv 3500  cdif 3937  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564   cuni 4837  {copab 5125  cmpt 5143  ccnv 5553  dom cdm 5554  ran crn 5555  cima 5557  Rel wrel 5559  Fun wfun 6346   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148   supp csupp 7821  Fincfn 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-ac2 9874
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-supp 7822  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-fin 8502  df-card 9357  df-acn 9360  df-ac 9531
This theorem is referenced by:  fpwrelmapffs  30383
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