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Theorem fin1a2lem11 9825
 Description: Lemma for fin1a2 9830. (Contributed by Stefan O'Rear, 8-Nov-2014.)
Assertion
Ref Expression
fin1a2lem11 (( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝐴 ∪ {∅}))
Distinct variable group:   𝑏,𝑐,𝐴

Proof of Theorem fin1a2lem11
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 eqid 2801 . . 3 (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏})
21rnmpt 5795 . 2 ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}}
3 unieq 4814 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
4 uni0 4831 . . . . . . . . . . . 12 ∅ = ∅
53, 4eqtrdi 2852 . . . . . . . . . . 11 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
65adantl 485 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → {𝑐𝐴𝑐𝑏} = ∅)
7 0ex 5178 . . . . . . . . . . 11 ∅ ∈ V
87elsn2 4567 . . . . . . . . . 10 ( {𝑐𝐴𝑐𝑏} ∈ {∅} ↔ {𝑐𝐴𝑐𝑏} = ∅)
96, 8sylibr 237 . . . . . . . . 9 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → {𝑐𝐴𝑐𝑏} ∈ {∅})
109olcd 871 . . . . . . . 8 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
11 ssrab2 4010 . . . . . . . . . 10 {𝑐𝐴𝑐𝑏} ⊆ 𝐴
12 simpr 488 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ≠ ∅)
13 fin1a2lem9 9823 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin ∧ 𝑏 ∈ ω) → {𝑐𝐴𝑐𝑏} ∈ Fin)
1413ad4ant123 1169 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ Fin)
15 simplll 774 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or 𝐴)
16 soss 5461 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} ⊆ 𝐴 → ( [] Or 𝐴 → [] Or {𝑐𝐴𝑐𝑏}))
1711, 15, 16mpsyl 68 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or {𝑐𝐴𝑐𝑏})
18 fin1a2lem10 9824 . . . . . . . . . . 11 (({𝑐𝐴𝑐𝑏} ≠ ∅ ∧ {𝑐𝐴𝑐𝑏} ∈ Fin ∧ [] Or {𝑐𝐴𝑐𝑏}) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
1912, 14, 17, 18syl3anc 1368 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
2011, 19sseldi 3916 . . . . . . . . 9 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ 𝐴)
2120orcd 870 . . . . . . . 8 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
2210, 21pm2.61dane 3077 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
23 eleq1 2880 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴 {𝑐𝐴𝑐𝑏} ∈ 𝐴))
24 eleq1 2880 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑 ∈ {∅} ↔ {𝑐𝐴𝑐𝑏} ∈ {∅}))
2523, 24orbi12d 916 . . . . . . 7 (𝑑 = {𝑐𝐴𝑐𝑏} → ((𝑑𝐴𝑑 ∈ {∅}) ↔ ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅})))
2622, 25syl5ibrcom 250 . . . . . 6 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
2726rexlimdva 3246 . . . . 5 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
28 simpr 488 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
2928sselda 3918 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ Fin)
30 ficardom 9378 . . . . . . . . 9 (𝑑 ∈ Fin → (card‘𝑑) ∈ ω)
3129, 30syl 17 . . . . . . . 8 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (card‘𝑑) ∈ ω)
32 breq1 5036 . . . . . . . . . . 11 (𝑐 = 𝑑 → (𝑐 ≼ (card‘𝑑) ↔ 𝑑 ≼ (card‘𝑑)))
33 simpr 488 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑𝐴)
34 ficardid 9379 . . . . . . . . . . . . 13 (𝑑 ∈ Fin → (card‘𝑑) ≈ 𝑑)
3529, 34syl 17 . . . . . . . . . . . 12 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (card‘𝑑) ≈ 𝑑)
36 ensym 8545 . . . . . . . . . . . 12 ((card‘𝑑) ≈ 𝑑𝑑 ≈ (card‘𝑑))
37 endom 8523 . . . . . . . . . . . 12 (𝑑 ≈ (card‘𝑑) → 𝑑 ≼ (card‘𝑑))
3835, 36, 373syl 18 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ≼ (card‘𝑑))
3932, 33, 38elrabd 3633 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)})
40 elssuni 4833 . . . . . . . . . 10 (𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
4139, 40syl 17 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
42 breq1 5036 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑐 ≼ (card‘𝑑) ↔ 𝑏 ≼ (card‘𝑑)))
4342elrab 3631 . . . . . . . . . . . 12 (𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} ↔ (𝑏𝐴𝑏 ≼ (card‘𝑑)))
44 simprr 772 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ (card‘𝑑))
4535adantr 484 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (card‘𝑑) ≈ 𝑑)
46 domentr 8555 . . . . . . . . . . . . . . 15 ((𝑏 ≼ (card‘𝑑) ∧ (card‘𝑑) ≈ 𝑑) → 𝑏𝑑)
4744, 45, 46syl2anc 587 . . . . . . . . . . . . . 14 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝑑)
48 simpllr 775 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝐴 ⊆ Fin)
49 simprl 770 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝐴)
5048, 49sseldd 3919 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ Fin)
5129adantr 484 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ Fin)
52 simplll 774 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → [] Or 𝐴)
53 simplr 768 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑𝐴)
54 sorpssi 7439 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑏𝐴𝑑𝐴)) → (𝑏𝑑𝑑𝑏))
5552, 49, 53, 54syl12anc 835 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (𝑏𝑑𝑑𝑏))
56 fincssdom 9738 . . . . . . . . . . . . . . 15 ((𝑏 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑏𝑑𝑑𝑏)) → (𝑏𝑑𝑏𝑑))
5750, 51, 55, 56syl3anc 1368 . . . . . . . . . . . . . 14 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (𝑏𝑑𝑏𝑑))
5847, 57mpbid 235 . . . . . . . . . . . . 13 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝑑)
5958ex 416 . . . . . . . . . . . 12 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ((𝑏𝐴𝑏 ≼ (card‘𝑑)) → 𝑏𝑑))
6043, 59syl5bi 245 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} → 𝑏𝑑))
6160ralrimiv 3151 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
62 unissb 4835 . . . . . . . . . 10 ( {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑 ↔ ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
6361, 62sylibr 237 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑)
6441, 63eqssd 3935 . . . . . . . 8 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
65 breq2 5037 . . . . . . . . . . 11 (𝑏 = (card‘𝑑) → (𝑐𝑏𝑐 ≼ (card‘𝑑)))
6665rabbidv 3430 . . . . . . . . . 10 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
6766unieqd 4817 . . . . . . . . 9 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
6867rspceeqv 3589 . . . . . . . 8 (((card‘𝑑) ∈ ω ∧ 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)}) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
6931, 64, 68syl2anc 587 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
7069ex 416 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑𝐴 → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
71 velsn 4544 . . . . . . 7 (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
72 peano1 7585 . . . . . . . . 9 ∅ ∈ ω
73 dom0 8633 . . . . . . . . . . . . . . . 16 (𝑏 ≼ ∅ ↔ 𝑏 = ∅)
7473biimpi 219 . . . . . . . . . . . . . . 15 (𝑏 ≼ ∅ → 𝑏 = ∅)
7574adantl 485 . . . . . . . . . . . . . 14 ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅)
7675a1i 11 . . . . . . . . . . . . 13 (( [] Or 𝐴𝐴 ⊆ Fin) → ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅))
77 breq1 5036 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (𝑐 ≼ ∅ ↔ 𝑏 ≼ ∅))
7877elrab 3631 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} ↔ (𝑏𝐴𝑏 ≼ ∅))
79 velsn 4544 . . . . . . . . . . . . 13 (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
8076, 78, 793imtr4g 299 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} → 𝑏 ∈ {∅}))
8180ssrdv 3924 . . . . . . . . . . 11 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
82 uni0b 4829 . . . . . . . . . . 11 ( {𝑐𝐴𝑐 ≼ ∅} = ∅ ↔ {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
8381, 82sylibr 237 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} = ∅)
8483eqcomd 2807 . . . . . . . . 9 (( [] Or 𝐴𝐴 ⊆ Fin) → ∅ = {𝑐𝐴𝑐 ≼ ∅})
85 breq2 5037 . . . . . . . . . . . 12 (𝑏 = ∅ → (𝑐𝑏𝑐 ≼ ∅))
8685rabbidv 3430 . . . . . . . . . . 11 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
8786unieqd 4817 . . . . . . . . . 10 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
8887rspceeqv 3589 . . . . . . . . 9 ((∅ ∈ ω ∧ ∅ = {𝑐𝐴𝑐 ≼ ∅}) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
8972, 84, 88sylancr 590 . . . . . . . 8 (( [] Or 𝐴𝐴 ⊆ Fin) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
90 eqeq1 2805 . . . . . . . . 9 (𝑑 = ∅ → (𝑑 = {𝑐𝐴𝑐𝑏} ↔ ∅ = {𝑐𝐴𝑐𝑏}))
9190rexbidv 3259 . . . . . . . 8 (𝑑 = ∅ → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏}))
9289, 91syl5ibrcom 250 . . . . . . 7 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑 = ∅ → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
9371, 92syl5bi 245 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑 ∈ {∅} → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
9470, 93jaod 856 . . . . 5 (( [] Or 𝐴𝐴 ⊆ Fin) → ((𝑑𝐴𝑑 ∈ {∅}) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
9527, 94impbid 215 . . . 4 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ (𝑑𝐴𝑑 ∈ {∅})))
96 elun 4079 . . . 4 (𝑑 ∈ (𝐴 ∪ {∅}) ↔ (𝑑𝐴𝑑 ∈ {∅}))
9795, 96syl6bbr 292 . . 3 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ 𝑑 ∈ (𝐴 ∪ {∅})))
9897abbi1dv 2931 . 2 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}} = (𝐴 ∪ {∅}))
992, 98syl5eq 2848 1 (( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝐴 ∪ {∅}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2112  {cab 2779   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  {crab 3113   ∪ cun 3882   ⊆ wss 3884  ∅c0 4246  {csn 4528  ∪ cuni 4803   class class class wbr 5033   ↦ cmpt 5113   Or wor 5441  ran crn 5524  ‘cfv 6328   [⊊] crpss 7432  ωcom 7564   ≈ cen 8493   ≼ cdom 8494  Fincfn 8496  cardccrd 9352 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-rpss 7433  df-om 7565  df-1o 8089  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356 This theorem is referenced by:  fin1a2lem12  9826
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