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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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