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Theorem fin1a2lem11 9176
Description: Lemma for fin1a2 9181. (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 2621 . . 3 (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏})
21rnmpt 5331 . 2 ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}}
3 unieq 4410 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
4 uni0 4431 . . . . . . . . . . . 12 ∅ = ∅
53, 4syl6eq 2671 . . . . . . . . . . 11 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
65adantl 482 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → {𝑐𝐴𝑐𝑏} = ∅)
7 0ex 4750 . . . . . . . . . . 11 ∅ ∈ V
87elsn2 4182 . . . . . . . . . 10 ( {𝑐𝐴𝑐𝑏} ∈ {∅} ↔ {𝑐𝐴𝑐𝑏} = ∅)
96, 8sylibr 224 . . . . . . . . 9 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → {𝑐𝐴𝑐𝑏} ∈ {∅})
109olcd 408 . . . . . . . 8 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
11 ssrab2 3666 . . . . . . . . . 10 {𝑐𝐴𝑐𝑏} ⊆ 𝐴
12 simpr 477 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ≠ ∅)
13 simplll 797 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or 𝐴)
14 simpllr 798 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → 𝐴 ⊆ Fin)
15 simplr 791 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → 𝑏 ∈ ω)
16 fin1a2lem9 9174 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin ∧ 𝑏 ∈ ω) → {𝑐𝐴𝑐𝑏} ∈ Fin)
1713, 14, 15, 16syl3anc 1323 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ Fin)
18 soss 5013 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} ⊆ 𝐴 → ( [] Or 𝐴 → [] Or {𝑐𝐴𝑐𝑏}))
1911, 13, 18mpsyl 68 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or {𝑐𝐴𝑐𝑏})
20 fin1a2lem10 9175 . . . . . . . . . . 11 (({𝑐𝐴𝑐𝑏} ≠ ∅ ∧ {𝑐𝐴𝑐𝑏} ∈ Fin ∧ [] Or {𝑐𝐴𝑐𝑏}) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
2112, 17, 19, 20syl3anc 1323 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
2211, 21sseldi 3581 . . . . . . . . 9 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ 𝐴)
2322orcd 407 . . . . . . . 8 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
2410, 23pm2.61dane 2877 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
25 eleq1 2686 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴 {𝑐𝐴𝑐𝑏} ∈ 𝐴))
26 eleq1 2686 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑 ∈ {∅} ↔ {𝑐𝐴𝑐𝑏} ∈ {∅}))
2725, 26orbi12d 745 . . . . . . 7 (𝑑 = {𝑐𝐴𝑐𝑏} → ((𝑑𝐴𝑑 ∈ {∅}) ↔ ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅})))
2824, 27syl5ibrcom 237 . . . . . 6 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
2928rexlimdva 3024 . . . . 5 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
30 simpr 477 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
3130sselda 3583 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ Fin)
32 ficardom 8731 . . . . . . . . 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 8732 . . . . . . . . . . . . 13 (𝑑 ∈ Fin → (card‘𝑑) ≈ 𝑑)
3631, 35syl 17 . . . . . . . . . . . 12 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (card‘𝑑) ≈ 𝑑)
37 ensym 7949 . . . . . . . . . . . 12 ((card‘𝑑) ≈ 𝑑𝑑 ≈ (card‘𝑑))
38 endom 7926 . . . . . . . . . . . 12 (𝑑 ≈ (card‘𝑑) → 𝑑 ≼ (card‘𝑑))
3936, 37, 383syl 18 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ≼ (card‘𝑑))
40 breq1 4616 . . . . . . . . . . . 12 (𝑐 = 𝑑 → (𝑐 ≼ (card‘𝑑) ↔ 𝑑 ≼ (card‘𝑑)))
4140elrab 3346 . . . . . . . . . . 11 (𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} ↔ (𝑑𝐴𝑑 ≼ (card‘𝑑)))
4234, 39, 41sylanbrc 697 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)})
43 elssuni 4433 . . . . . . . . . 10 (𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
4442, 43syl 17 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
45 breq1 4616 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑐 ≼ (card‘𝑑) ↔ 𝑏 ≼ (card‘𝑑)))
4645elrab 3346 . . . . . . . . . . . 12 (𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} ↔ (𝑏𝐴𝑏 ≼ (card‘𝑑)))
47 simprr 795 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ (card‘𝑑))
4836adantr 481 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (card‘𝑑) ≈ 𝑑)
49 domentr 7959 . . . . . . . . . . . . . . 15 ((𝑏 ≼ (card‘𝑑) ∧ (card‘𝑑) ≈ 𝑑) → 𝑏𝑑)
5047, 48, 49syl2anc 692 . . . . . . . . . . . . . 14 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝑑)
51 simpllr 798 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝐴 ⊆ Fin)
52 simprl 793 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝐴)
5351, 52sseldd 3584 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ Fin)
5431adantr 481 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ Fin)
55 simplll 797 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → [] Or 𝐴)
56 simplr 791 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑𝐴)
57 sorpssi 6896 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑏𝐴𝑑𝐴)) → (𝑏𝑑𝑑𝑏))
5855, 52, 56, 57syl12anc 1321 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (𝑏𝑑𝑑𝑏))
59 fincssdom 9089 . . . . . . . . . . . . . . 15 ((𝑏 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑏𝑑𝑑𝑏)) → (𝑏𝑑𝑏𝑑))
6053, 54, 58, 59syl3anc 1323 . . . . . . . . . . . . . 14 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (𝑏𝑑𝑏𝑑))
6150, 60mpbid 222 . . . . . . . . . . . . 13 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝑑)
6261ex 450 . . . . . . . . . . . 12 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ((𝑏𝐴𝑏 ≼ (card‘𝑑)) → 𝑏𝑑))
6346, 62syl5bi 232 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} → 𝑏𝑑))
6463ralrimiv 2959 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
65 unissb 4435 . . . . . . . . . 10 ( {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑 ↔ ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
6664, 65sylibr 224 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑)
6744, 66eqssd 3600 . . . . . . . 8 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
68 breq2 4617 . . . . . . . . . . . 12 (𝑏 = (card‘𝑑) → (𝑐𝑏𝑐 ≼ (card‘𝑑)))
6968rabbidv 3177 . . . . . . . . . . 11 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
7069unieqd 4412 . . . . . . . . . 10 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
7170eqeq2d 2631 . . . . . . . . 9 (𝑏 = (card‘𝑑) → (𝑑 = {𝑐𝐴𝑐𝑏} ↔ 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)}))
7271rspcev 3295 . . . . . . . 8 (((card‘𝑑) ∈ ω ∧ 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)}) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
7333, 67, 72syl2anc 692 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
7473ex 450 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑𝐴 → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
75 velsn 4164 . . . . . . 7 (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
76 peano1 7032 . . . . . . . . 9 ∅ ∈ ω
77 dom0 8032 . . . . . . . . . . . . . . . 16 (𝑏 ≼ ∅ ↔ 𝑏 = ∅)
7877biimpi 206 . . . . . . . . . . . . . . 15 (𝑏 ≼ ∅ → 𝑏 = ∅)
7978adantl 482 . . . . . . . . . . . . . 14 ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅)
8079a1i 11 . . . . . . . . . . . . 13 (( [] Or 𝐴𝐴 ⊆ Fin) → ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅))
81 breq1 4616 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (𝑐 ≼ ∅ ↔ 𝑏 ≼ ∅))
8281elrab 3346 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} ↔ (𝑏𝐴𝑏 ≼ ∅))
83 velsn 4164 . . . . . . . . . . . . 13 (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
8480, 82, 833imtr4g 285 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} → 𝑏 ∈ {∅}))
8584ssrdv 3589 . . . . . . . . . . 11 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
86 uni0b 4429 . . . . . . . . . . 11 ( {𝑐𝐴𝑐 ≼ ∅} = ∅ ↔ {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
8785, 86sylibr 224 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} = ∅)
8887eqcomd 2627 . . . . . . . . 9 (( [] Or 𝐴𝐴 ⊆ Fin) → ∅ = {𝑐𝐴𝑐 ≼ ∅})
89 breq2 4617 . . . . . . . . . . . . 13 (𝑏 = ∅ → (𝑐𝑏𝑐 ≼ ∅))
9089rabbidv 3177 . . . . . . . . . . . 12 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
9190unieqd 4412 . . . . . . . . . . 11 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
9291eqeq2d 2631 . . . . . . . . . 10 (𝑏 = ∅ → (∅ = {𝑐𝐴𝑐𝑏} ↔ ∅ = {𝑐𝐴𝑐 ≼ ∅}))
9392rspcev 3295 . . . . . . . . 9 ((∅ ∈ ω ∧ ∅ = {𝑐𝐴𝑐 ≼ ∅}) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
9476, 88, 93sylancr 694 . . . . . . . 8 (( [] Or 𝐴𝐴 ⊆ Fin) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
95 eqeq1 2625 . . . . . . . . 9 (𝑑 = ∅ → (𝑑 = {𝑐𝐴𝑐𝑏} ↔ ∅ = {𝑐𝐴𝑐𝑏}))
9695rexbidv 3045 . . . . . . . 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 395 . . . . 5 (( [] Or 𝐴𝐴 ⊆ Fin) → ((𝑑𝐴𝑑 ∈ {∅}) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
10029, 99impbid 202 . . . 4 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ (𝑑𝐴𝑑 ∈ {∅})))
101 elun 3731 . . . 4 (𝑑 ∈ (𝐴 ∪ {∅}) ↔ (𝑑𝐴𝑑 ∈ {∅}))
102100, 101syl6bbr 278 . . 3 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ 𝑑 ∈ (𝐴 ∪ {∅})))
103102abbi1dv 2740 . 2 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}} = (𝐴 ∪ {∅}))
1042, 103syl5eq 2667 1 (( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝐴 ∪ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  cun 3553  wss 3555  c0 3891  {csn 4148   cuni 4402   class class class wbr 4613  cmpt 4673   Or wor 4994  ran crn 5075  cfv 5847   [] crpss 6889  ωcom 7012  cen 7896  cdom 7897  Fincfn 7899  cardccrd 8705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-rpss 6890  df-om 7013  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709
This theorem is referenced by:  fin1a2lem12  9177
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