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Theorem fin1a2lem11 9485
Description: Lemma for fin1a2 9490. (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 2765 . . 3 (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏})
21rnmpt 5540 . 2 ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}}
3 unieq 4602 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
4 uni0 4623 . . . . . . . . . . . 12 ∅ = ∅
53, 4syl6eq 2815 . . . . . . . . . . 11 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
65adantl 473 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → {𝑐𝐴𝑐𝑏} = ∅)
7 0ex 4950 . . . . . . . . . . 11 ∅ ∈ V
87elsn2 4369 . . . . . . . . . 10 ( {𝑐𝐴𝑐𝑏} ∈ {∅} ↔ {𝑐𝐴𝑐𝑏} = ∅)
96, 8sylibr 225 . . . . . . . . 9 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → {𝑐𝐴𝑐𝑏} ∈ {∅})
109olcd 900 . . . . . . . 8 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
11 ssrab2 3847 . . . . . . . . . 10 {𝑐𝐴𝑐𝑏} ⊆ 𝐴
12 simpr 477 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ≠ ∅)
13 simplll 791 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or 𝐴)
14 simpllr 793 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → 𝐴 ⊆ Fin)
15 simplr 785 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → 𝑏 ∈ ω)
16 fin1a2lem9 9483 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin ∧ 𝑏 ∈ ω) → {𝑐𝐴𝑐𝑏} ∈ Fin)
1713, 14, 15, 16syl3anc 1490 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ Fin)
18 soss 5216 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} ⊆ 𝐴 → ( [] Or 𝐴 → [] Or {𝑐𝐴𝑐𝑏}))
1911, 13, 18mpsyl 68 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or {𝑐𝐴𝑐𝑏})
20 fin1a2lem10 9484 . . . . . . . . . . 11 (({𝑐𝐴𝑐𝑏} ≠ ∅ ∧ {𝑐𝐴𝑐𝑏} ∈ Fin ∧ [] Or {𝑐𝐴𝑐𝑏}) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
2112, 17, 19, 20syl3anc 1490 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
2211, 21sseldi 3759 . . . . . . . . 9 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ 𝐴)
2322orcd 899 . . . . . . . 8 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
2410, 23pm2.61dane 3024 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
25 eleq1 2832 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴 {𝑐𝐴𝑐𝑏} ∈ 𝐴))
26 eleq1 2832 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑 ∈ {∅} ↔ {𝑐𝐴𝑐𝑏} ∈ {∅}))
2725, 26orbi12d 942 . . . . . . 7 (𝑑 = {𝑐𝐴𝑐𝑏} → ((𝑑𝐴𝑑 ∈ {∅}) ↔ ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅})))
2824, 27syl5ibrcom 238 . . . . . 6 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
2928rexlimdva 3178 . . . . 5 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
30 simpr 477 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
3130sselda 3761 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ Fin)
32 ficardom 9038 . . . . . . . . 9 (𝑑 ∈ Fin → (card‘𝑑) ∈ ω)
3331, 32syl 17 . . . . . . . 8 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (card‘𝑑) ∈ ω)
34 simpr 477 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑𝐴)
35 ficardid 9039 . . . . . . . . . . . . 13 (𝑑 ∈ Fin → (card‘𝑑) ≈ 𝑑)
3631, 35syl 17 . . . . . . . . . . . 12 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (card‘𝑑) ≈ 𝑑)
37 ensym 8209 . . . . . . . . . . . 12 ((card‘𝑑) ≈ 𝑑𝑑 ≈ (card‘𝑑))
38 endom 8187 . . . . . . . . . . . 12 (𝑑 ≈ (card‘𝑑) → 𝑑 ≼ (card‘𝑑))
3936, 37, 383syl 18 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ≼ (card‘𝑑))
40 breq1 4812 . . . . . . . . . . . 12 (𝑐 = 𝑑 → (𝑐 ≼ (card‘𝑑) ↔ 𝑑 ≼ (card‘𝑑)))
4140elrab 3519 . . . . . . . . . . 11 (𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} ↔ (𝑑𝐴𝑑 ≼ (card‘𝑑)))
4234, 39, 41sylanbrc 578 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)})
43 elssuni 4625 . . . . . . . . . 10 (𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
4442, 43syl 17 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
45 breq1 4812 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑐 ≼ (card‘𝑑) ↔ 𝑏 ≼ (card‘𝑑)))
4645elrab 3519 . . . . . . . . . . . 12 (𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} ↔ (𝑏𝐴𝑏 ≼ (card‘𝑑)))
47 simprr 789 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ (card‘𝑑))
4836adantr 472 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (card‘𝑑) ≈ 𝑑)
49 domentr 8219 . . . . . . . . . . . . . . 15 ((𝑏 ≼ (card‘𝑑) ∧ (card‘𝑑) ≈ 𝑑) → 𝑏𝑑)
5047, 48, 49syl2anc 579 . . . . . . . . . . . . . 14 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝑑)
51 simpllr 793 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝐴 ⊆ Fin)
52 simprl 787 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝐴)
5351, 52sseldd 3762 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ Fin)
5431adantr 472 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ Fin)
55 simplll 791 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → [] Or 𝐴)
56 simplr 785 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑𝐴)
57 sorpssi 7141 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑏𝐴𝑑𝐴)) → (𝑏𝑑𝑑𝑏))
5855, 52, 56, 57syl12anc 865 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (𝑏𝑑𝑑𝑏))
59 fincssdom 9398 . . . . . . . . . . . . . . 15 ((𝑏 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑏𝑑𝑑𝑏)) → (𝑏𝑑𝑏𝑑))
6053, 54, 58, 59syl3anc 1490 . . . . . . . . . . . . . 14 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (𝑏𝑑𝑏𝑑))
6150, 60mpbid 223 . . . . . . . . . . . . 13 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝑑)
6261ex 401 . . . . . . . . . . . 12 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ((𝑏𝐴𝑏 ≼ (card‘𝑑)) → 𝑏𝑑))
6346, 62syl5bi 233 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} → 𝑏𝑑))
6463ralrimiv 3112 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
65 unissb 4627 . . . . . . . . . 10 ( {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑 ↔ ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
6664, 65sylibr 225 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑)
6744, 66eqssd 3778 . . . . . . . 8 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
68 breq2 4813 . . . . . . . . . . 11 (𝑏 = (card‘𝑑) → (𝑐𝑏𝑐 ≼ (card‘𝑑)))
6968rabbidv 3338 . . . . . . . . . 10 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
7069unieqd 4604 . . . . . . . . 9 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
7170rspceeqv 3479 . . . . . . . 8 (((card‘𝑑) ∈ ω ∧ 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)}) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
7233, 67, 71syl2anc 579 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
7372ex 401 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑𝐴 → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
74 velsn 4350 . . . . . . 7 (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
75 peano1 7283 . . . . . . . . 9 ∅ ∈ ω
76 dom0 8295 . . . . . . . . . . . . . . . 16 (𝑏 ≼ ∅ ↔ 𝑏 = ∅)
7776biimpi 207 . . . . . . . . . . . . . . 15 (𝑏 ≼ ∅ → 𝑏 = ∅)
7877adantl 473 . . . . . . . . . . . . . 14 ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅)
7978a1i 11 . . . . . . . . . . . . 13 (( [] Or 𝐴𝐴 ⊆ Fin) → ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅))
80 breq1 4812 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (𝑐 ≼ ∅ ↔ 𝑏 ≼ ∅))
8180elrab 3519 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} ↔ (𝑏𝐴𝑏 ≼ ∅))
82 velsn 4350 . . . . . . . . . . . . 13 (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
8379, 81, 823imtr4g 287 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} → 𝑏 ∈ {∅}))
8483ssrdv 3767 . . . . . . . . . . 11 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
85 uni0b 4621 . . . . . . . . . . 11 ( {𝑐𝐴𝑐 ≼ ∅} = ∅ ↔ {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
8684, 85sylibr 225 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} = ∅)
8786eqcomd 2771 . . . . . . . . 9 (( [] Or 𝐴𝐴 ⊆ Fin) → ∅ = {𝑐𝐴𝑐 ≼ ∅})
88 breq2 4813 . . . . . . . . . . . 12 (𝑏 = ∅ → (𝑐𝑏𝑐 ≼ ∅))
8988rabbidv 3338 . . . . . . . . . . 11 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
9089unieqd 4604 . . . . . . . . . 10 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
9190rspceeqv 3479 . . . . . . . . 9 ((∅ ∈ ω ∧ ∅ = {𝑐𝐴𝑐 ≼ ∅}) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
9275, 87, 91sylancr 581 . . . . . . . 8 (( [] Or 𝐴𝐴 ⊆ Fin) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
93 eqeq1 2769 . . . . . . . . 9 (𝑑 = ∅ → (𝑑 = {𝑐𝐴𝑐𝑏} ↔ ∅ = {𝑐𝐴𝑐𝑏}))
9493rexbidv 3199 . . . . . . . 8 (𝑑 = ∅ → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏}))
9592, 94syl5ibrcom 238 . . . . . . 7 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑 = ∅ → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
9674, 95syl5bi 233 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑 ∈ {∅} → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
9773, 96jaod 885 . . . . 5 (( [] Or 𝐴𝐴 ⊆ Fin) → ((𝑑𝐴𝑑 ∈ {∅}) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
9829, 97impbid 203 . . . 4 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ (𝑑𝐴𝑑 ∈ {∅})))
99 elun 3915 . . . 4 (𝑑 ∈ (𝐴 ∪ {∅}) ↔ (𝑑𝐴𝑑 ∈ {∅}))
10098, 99syl6bbr 280 . . 3 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ 𝑑 ∈ (𝐴 ∪ {∅})))
101100abbi1dv 2886 . 2 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}} = (𝐴 ∪ {∅}))
1022, 101syl5eq 2811 1 (( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝐴 ∪ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  cun 3730  wss 3732  c0 4079  {csn 4334   cuni 4594   class class class wbr 4809  cmpt 4888   Or wor 5197  ran crn 5278  cfv 6068   [] crpss 7134  ωcom 7263  cen 8157  cdom 8158  Fincfn 8160  cardccrd 9012
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 2070  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
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 2063  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-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-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-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-rpss 7135  df-om 7264  df-1o 7764  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016
This theorem is referenced by:  fin1a2lem12  9486
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