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Theorem fin1a2lem11 9444
Description: Lemma for fin1a2 9449. (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 2760 . . 3 (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏})
21rnmpt 5526 . 2 ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}}
3 unieq 4596 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
4 uni0 4617 . . . . . . . . . . . 12 ∅ = ∅
53, 4syl6eq 2810 . . . . . . . . . . 11 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
65adantl 473 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → {𝑐𝐴𝑐𝑏} = ∅)
7 0ex 4942 . . . . . . . . . . 11 ∅ ∈ V
87elsn2 4356 . . . . . . . . . 10 ( {𝑐𝐴𝑐𝑏} ∈ {∅} ↔ {𝑐𝐴𝑐𝑏} = ∅)
96, 8sylibr 224 . . . . . . . . 9 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → {𝑐𝐴𝑐𝑏} ∈ {∅})
109olcd 407 . . . . . . . 8 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
11 ssrab2 3828 . . . . . . . . . 10 {𝑐𝐴𝑐𝑏} ⊆ 𝐴
12 simpr 479 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ≠ ∅)
13 simplll 815 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or 𝐴)
14 simpllr 817 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → 𝐴 ⊆ Fin)
15 simplr 809 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → 𝑏 ∈ ω)
16 fin1a2lem9 9442 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin ∧ 𝑏 ∈ ω) → {𝑐𝐴𝑐𝑏} ∈ Fin)
1713, 14, 15, 16syl3anc 1477 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ Fin)
18 soss 5205 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} ⊆ 𝐴 → ( [] Or 𝐴 → [] Or {𝑐𝐴𝑐𝑏}))
1911, 13, 18mpsyl 68 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or {𝑐𝐴𝑐𝑏})
20 fin1a2lem10 9443 . . . . . . . . . . 11 (({𝑐𝐴𝑐𝑏} ≠ ∅ ∧ {𝑐𝐴𝑐𝑏} ∈ Fin ∧ [] Or {𝑐𝐴𝑐𝑏}) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
2112, 17, 19, 20syl3anc 1477 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
2211, 21sseldi 3742 . . . . . . . . 9 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ 𝐴)
2322orcd 406 . . . . . . . 8 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
2410, 23pm2.61dane 3019 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
25 eleq1 2827 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴 {𝑐𝐴𝑐𝑏} ∈ 𝐴))
26 eleq1 2827 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑 ∈ {∅} ↔ {𝑐𝐴𝑐𝑏} ∈ {∅}))
2725, 26orbi12d 748 . . . . . . 7 (𝑑 = {𝑐𝐴𝑐𝑏} → ((𝑑𝐴𝑑 ∈ {∅}) ↔ ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅})))
2824, 27syl5ibrcom 237 . . . . . 6 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
2928rexlimdva 3169 . . . . 5 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
30 simpr 479 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
3130sselda 3744 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ Fin)
32 ficardom 8997 . . . . . . . . 9 (𝑑 ∈ Fin → (card‘𝑑) ∈ ω)
3331, 32syl 17 . . . . . . . 8 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (card‘𝑑) ∈ ω)
34 simpr 479 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑𝐴)
35 ficardid 8998 . . . . . . . . . . . . 13 (𝑑 ∈ Fin → (card‘𝑑) ≈ 𝑑)
3631, 35syl 17 . . . . . . . . . . . 12 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (card‘𝑑) ≈ 𝑑)
37 ensym 8172 . . . . . . . . . . . 12 ((card‘𝑑) ≈ 𝑑𝑑 ≈ (card‘𝑑))
38 endom 8150 . . . . . . . . . . . 12 (𝑑 ≈ (card‘𝑑) → 𝑑 ≼ (card‘𝑑))
3936, 37, 383syl 18 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ≼ (card‘𝑑))
40 breq1 4807 . . . . . . . . . . . 12 (𝑐 = 𝑑 → (𝑐 ≼ (card‘𝑑) ↔ 𝑑 ≼ (card‘𝑑)))
4140elrab 3504 . . . . . . . . . . 11 (𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} ↔ (𝑑𝐴𝑑 ≼ (card‘𝑑)))
4234, 39, 41sylanbrc 701 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)})
43 elssuni 4619 . . . . . . . . . 10 (𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
4442, 43syl 17 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
45 breq1 4807 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑐 ≼ (card‘𝑑) ↔ 𝑏 ≼ (card‘𝑑)))
4645elrab 3504 . . . . . . . . . . . 12 (𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} ↔ (𝑏𝐴𝑏 ≼ (card‘𝑑)))
47 simprr 813 . . . . . . . . . . . . . . 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 8182 . . . . . . . . . . . . . . 15 ((𝑏 ≼ (card‘𝑑) ∧ (card‘𝑑) ≈ 𝑑) → 𝑏𝑑)
5047, 48, 49syl2anc 696 . . . . . . . . . . . . . 14 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝑑)
51 simpllr 817 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝐴 ⊆ Fin)
52 simprl 811 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝐴)
5351, 52sseldd 3745 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ Fin)
5431adantr 472 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ Fin)
55 simplll 815 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → [] Or 𝐴)
56 simplr 809 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑𝐴)
57 sorpssi 7109 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑏𝐴𝑑𝐴)) → (𝑏𝑑𝑑𝑏))
5855, 52, 56, 57syl12anc 1475 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (𝑏𝑑𝑑𝑏))
59 fincssdom 9357 . . . . . . . . . . . . . . 15 ((𝑏 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑏𝑑𝑑𝑏)) → (𝑏𝑑𝑏𝑑))
6053, 54, 58, 59syl3anc 1477 . . . . . . . . . . . . . 14 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (𝑏𝑑𝑏𝑑))
6150, 60mpbid 222 . . . . . . . . . . . . 13 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝑑)
6261ex 449 . . . . . . . . . . . 12 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ((𝑏𝐴𝑏 ≼ (card‘𝑑)) → 𝑏𝑑))
6346, 62syl5bi 232 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} → 𝑏𝑑))
6463ralrimiv 3103 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
65 unissb 4621 . . . . . . . . . 10 ( {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑 ↔ ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
6664, 65sylibr 224 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑)
6744, 66eqssd 3761 . . . . . . . 8 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
68 breq2 4808 . . . . . . . . . . . 12 (𝑏 = (card‘𝑑) → (𝑐𝑏𝑐 ≼ (card‘𝑑)))
6968rabbidv 3329 . . . . . . . . . . 11 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
7069unieqd 4598 . . . . . . . . . 10 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
7170eqeq2d 2770 . . . . . . . . 9 (𝑏 = (card‘𝑑) → (𝑑 = {𝑐𝐴𝑐𝑏} ↔ 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)}))
7271rspcev 3449 . . . . . . . 8 (((card‘𝑑) ∈ ω ∧ 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)}) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
7333, 67, 72syl2anc 696 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
7473ex 449 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑𝐴 → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
75 velsn 4337 . . . . . . 7 (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
76 peano1 7251 . . . . . . . . 9 ∅ ∈ ω
77 dom0 8255 . . . . . . . . . . . . . . . 16 (𝑏 ≼ ∅ ↔ 𝑏 = ∅)
7877biimpi 206 . . . . . . . . . . . . . . 15 (𝑏 ≼ ∅ → 𝑏 = ∅)
7978adantl 473 . . . . . . . . . . . . . 14 ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅)
8079a1i 11 . . . . . . . . . . . . 13 (( [] Or 𝐴𝐴 ⊆ Fin) → ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅))
81 breq1 4807 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (𝑐 ≼ ∅ ↔ 𝑏 ≼ ∅))
8281elrab 3504 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} ↔ (𝑏𝐴𝑏 ≼ ∅))
83 velsn 4337 . . . . . . . . . . . . 13 (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
8480, 82, 833imtr4g 285 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} → 𝑏 ∈ {∅}))
8584ssrdv 3750 . . . . . . . . . . 11 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
86 uni0b 4615 . . . . . . . . . . 11 ( {𝑐𝐴𝑐 ≼ ∅} = ∅ ↔ {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
8785, 86sylibr 224 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} = ∅)
8887eqcomd 2766 . . . . . . . . 9 (( [] Or 𝐴𝐴 ⊆ Fin) → ∅ = {𝑐𝐴𝑐 ≼ ∅})
89 breq2 4808 . . . . . . . . . . . . 13 (𝑏 = ∅ → (𝑐𝑏𝑐 ≼ ∅))
9089rabbidv 3329 . . . . . . . . . . . 12 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
9190unieqd 4598 . . . . . . . . . . 11 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
9291eqeq2d 2770 . . . . . . . . . 10 (𝑏 = ∅ → (∅ = {𝑐𝐴𝑐𝑏} ↔ ∅ = {𝑐𝐴𝑐 ≼ ∅}))
9392rspcev 3449 . . . . . . . . 9 ((∅ ∈ ω ∧ ∅ = {𝑐𝐴𝑐 ≼ ∅}) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
9476, 88, 93sylancr 698 . . . . . . . 8 (( [] Or 𝐴𝐴 ⊆ Fin) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
95 eqeq1 2764 . . . . . . . . 9 (𝑑 = ∅ → (𝑑 = {𝑐𝐴𝑐𝑏} ↔ ∅ = {𝑐𝐴𝑐𝑏}))
9695rexbidv 3190 . . . . . . . 8 (𝑑 = ∅ → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏}))
9794, 96syl5ibrcom 237 . . . . . . 7 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑 = ∅ → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
9875, 97syl5bi 232 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑 ∈ {∅} → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
9974, 98jaod 394 . . . . 5 (( [] Or 𝐴𝐴 ⊆ Fin) → ((𝑑𝐴𝑑 ∈ {∅}) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
10029, 99impbid 202 . . . 4 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ (𝑑𝐴𝑑 ∈ {∅})))
101 elun 3896 . . . 4 (𝑑 ∈ (𝐴 ∪ {∅}) ↔ (𝑑𝐴𝑑 ∈ {∅}))
102100, 101syl6bbr 278 . . 3 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ 𝑑 ∈ (𝐴 ∪ {∅})))
103102abbi1dv 2881 . 2 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}} = (𝐴 ∪ {∅}))
1042, 103syl5eq 2806 1 (( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝐴 ∪ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  {crab 3054  cun 3713  wss 3715  c0 4058  {csn 4321   cuni 4588   class class class wbr 4804  cmpt 4881   Or wor 5186  ran crn 5267  cfv 6049   [] crpss 7102  ωcom 7231  cen 8120  cdom 8121  Fincfn 8123  cardccrd 8971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-rpss 7103  df-om 7232  df-1o 7730  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975
This theorem is referenced by:  fin1a2lem12  9445
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