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Theorem fin1a2lem11 9270
 Description: Lemma for fin1a2 9275. (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 2651 . . 3 (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏})
21rnmpt 5403 . 2 ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}}
3 unieq 4476 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
4 uni0 4497 . . . . . . . . . . . 12 ∅ = ∅
53, 4syl6eq 2701 . . . . . . . . . . 11 ({𝑐𝐴𝑐𝑏} = ∅ → {𝑐𝐴𝑐𝑏} = ∅)
65adantl 481 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} = ∅) → {𝑐𝐴𝑐𝑏} = ∅)
7 0ex 4823 . . . . . . . . . . 11 ∅ ∈ V
87elsn2 4244 . . . . . . . . . 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 3720 . . . . . . . . . 10 {𝑐𝐴𝑐𝑏} ⊆ 𝐴
12 simpr 476 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ≠ ∅)
13 simplll 813 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or 𝐴)
14 simpllr 815 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → 𝐴 ⊆ Fin)
15 simplr 807 . . . . . . . . . . . 12 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → 𝑏 ∈ ω)
16 fin1a2lem9 9268 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin ∧ 𝑏 ∈ ω) → {𝑐𝐴𝑐𝑏} ∈ Fin)
1713, 14, 15, 16syl3anc 1366 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ Fin)
18 soss 5082 . . . . . . . . . . . 12 ({𝑐𝐴𝑐𝑏} ⊆ 𝐴 → ( [] Or 𝐴 → [] Or {𝑐𝐴𝑐𝑏}))
1911, 13, 18mpsyl 68 . . . . . . . . . . 11 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → [] Or {𝑐𝐴𝑐𝑏})
20 fin1a2lem10 9269 . . . . . . . . . . 11 (({𝑐𝐴𝑐𝑏} ≠ ∅ ∧ {𝑐𝐴𝑐𝑏} ∈ Fin ∧ [] Or {𝑐𝐴𝑐𝑏}) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
2112, 17, 19, 20syl3anc 1366 . . . . . . . . . 10 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ {𝑐𝐴𝑐𝑏})
2211, 21sseldi 3634 . . . . . . . . 9 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → {𝑐𝐴𝑐𝑏} ∈ 𝐴)
2322orcd 406 . . . . . . . 8 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐𝐴𝑐𝑏} ≠ ∅) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
2410, 23pm2.61dane 2910 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅}))
25 eleq1 2718 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴 {𝑐𝐴𝑐𝑏} ∈ 𝐴))
26 eleq1 2718 . . . . . . . 8 (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑 ∈ {∅} ↔ {𝑐𝐴𝑐𝑏} ∈ {∅}))
2725, 26orbi12d 746 . . . . . . 7 (𝑑 = {𝑐𝐴𝑐𝑏} → ((𝑑𝐴𝑑 ∈ {∅}) ↔ ( {𝑐𝐴𝑐𝑏} ∈ 𝐴 {𝑐𝐴𝑐𝑏} ∈ {∅})))
2824, 27syl5ibrcom 237 . . . . . 6 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
2928rexlimdva 3060 . . . . 5 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} → (𝑑𝐴𝑑 ∈ {∅})))
30 simpr 476 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
3130sselda 3636 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ Fin)
32 ficardom 8825 . . . . . . . . 9 (𝑑 ∈ Fin → (card‘𝑑) ∈ ω)
3331, 32syl 17 . . . . . . . 8 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (card‘𝑑) ∈ ω)
34 simpr 476 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑𝐴)
35 ficardid 8826 . . . . . . . . . . . . 13 (𝑑 ∈ Fin → (card‘𝑑) ≈ 𝑑)
3631, 35syl 17 . . . . . . . . . . . 12 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → (card‘𝑑) ≈ 𝑑)
37 ensym 8046 . . . . . . . . . . . 12 ((card‘𝑑) ≈ 𝑑𝑑 ≈ (card‘𝑑))
38 endom 8024 . . . . . . . . . . . 12 (𝑑 ≈ (card‘𝑑) → 𝑑 ≼ (card‘𝑑))
3936, 37, 383syl 18 . . . . . . . . . . 11 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ≼ (card‘𝑑))
40 breq1 4688 . . . . . . . . . . . 12 (𝑐 = 𝑑 → (𝑐 ≼ (card‘𝑑) ↔ 𝑑 ≼ (card‘𝑑)))
4140elrab 3396 . . . . . . . . . . 11 (𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} ↔ (𝑑𝐴𝑑 ≼ (card‘𝑑)))
4234, 39, 41sylanbrc 699 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)})
43 elssuni 4499 . . . . . . . . . 10 (𝑑 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
4442, 43syl 17 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 {𝑐𝐴𝑐 ≼ (card‘𝑑)})
45 breq1 4688 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (𝑐 ≼ (card‘𝑑) ↔ 𝑏 ≼ (card‘𝑑)))
4645elrab 3396 . . . . . . . . . . . 12 (𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)} ↔ (𝑏𝐴𝑏 ≼ (card‘𝑑)))
47 simprr 811 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ (card‘𝑑))
4836adantr 480 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (card‘𝑑) ≈ 𝑑)
49 domentr 8056 . . . . . . . . . . . . . . 15 ((𝑏 ≼ (card‘𝑑) ∧ (card‘𝑑) ≈ 𝑑) → 𝑏𝑑)
5047, 48, 49syl2anc 694 . . . . . . . . . . . . . 14 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝑑)
51 simpllr 815 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝐴 ⊆ Fin)
52 simprl 809 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏𝐴)
5351, 52sseldd 3637 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ Fin)
5431adantr 480 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ Fin)
55 simplll 813 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → [] Or 𝐴)
56 simplr 807 . . . . . . . . . . . . . . . 16 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → 𝑑𝐴)
57 sorpssi 6985 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑏𝐴𝑑𝐴)) → (𝑏𝑑𝑑𝑏))
5855, 52, 56, 57syl12anc 1364 . . . . . . . . . . . . . . 15 (((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) ∧ (𝑏𝐴𝑏 ≼ (card‘𝑑))) → (𝑏𝑑𝑑𝑏))
59 fincssdom 9183 . . . . . . . . . . . . . . 15 ((𝑏 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑏𝑑𝑑𝑏)) → (𝑏𝑑𝑏𝑑))
6053, 54, 58, 59syl3anc 1366 . . . . . . . . . . . . . 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 2994 . . . . . . . . . 10 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
65 unissb 4501 . . . . . . . . . 10 ( {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑 ↔ ∀𝑏 ∈ {𝑐𝐴𝑐 ≼ (card‘𝑑)}𝑏𝑑)
6664, 65sylibr 224 . . . . . . . . 9 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → {𝑐𝐴𝑐 ≼ (card‘𝑑)} ⊆ 𝑑)
6744, 66eqssd 3653 . . . . . . . 8 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
68 breq2 4689 . . . . . . . . . . . 12 (𝑏 = (card‘𝑑) → (𝑐𝑏𝑐 ≼ (card‘𝑑)))
6968rabbidv 3220 . . . . . . . . . . 11 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
7069unieqd 4478 . . . . . . . . . 10 (𝑏 = (card‘𝑑) → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ (card‘𝑑)})
7170eqeq2d 2661 . . . . . . . . 9 (𝑏 = (card‘𝑑) → (𝑑 = {𝑐𝐴𝑐𝑏} ↔ 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)}))
7271rspcev 3340 . . . . . . . 8 (((card‘𝑑) ∈ ω ∧ 𝑑 = {𝑐𝐴𝑐 ≼ (card‘𝑑)}) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
7333, 67, 72syl2anc 694 . . . . . . 7 ((( [] Or 𝐴𝐴 ⊆ Fin) ∧ 𝑑𝐴) → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏})
7473ex 449 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑑𝐴 → ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}))
75 velsn 4226 . . . . . . 7 (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
76 peano1 7127 . . . . . . . . 9 ∅ ∈ ω
77 dom0 8129 . . . . . . . . . . . . . . . 16 (𝑏 ≼ ∅ ↔ 𝑏 = ∅)
7877biimpi 206 . . . . . . . . . . . . . . 15 (𝑏 ≼ ∅ → 𝑏 = ∅)
7978adantl 481 . . . . . . . . . . . . . 14 ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅)
8079a1i 11 . . . . . . . . . . . . 13 (( [] Or 𝐴𝐴 ⊆ Fin) → ((𝑏𝐴𝑏 ≼ ∅) → 𝑏 = ∅))
81 breq1 4688 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (𝑐 ≼ ∅ ↔ 𝑏 ≼ ∅))
8281elrab 3396 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} ↔ (𝑏𝐴𝑏 ≼ ∅))
83 velsn 4226 . . . . . . . . . . . . 13 (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
8480, 82, 833imtr4g 285 . . . . . . . . . . . 12 (( [] Or 𝐴𝐴 ⊆ Fin) → (𝑏 ∈ {𝑐𝐴𝑐 ≼ ∅} → 𝑏 ∈ {∅}))
8584ssrdv 3642 . . . . . . . . . . 11 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
86 uni0b 4495 . . . . . . . . . . 11 ( {𝑐𝐴𝑐 ≼ ∅} = ∅ ↔ {𝑐𝐴𝑐 ≼ ∅} ⊆ {∅})
8785, 86sylibr 224 . . . . . . . . . 10 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑐𝐴𝑐 ≼ ∅} = ∅)
8887eqcomd 2657 . . . . . . . . 9 (( [] Or 𝐴𝐴 ⊆ Fin) → ∅ = {𝑐𝐴𝑐 ≼ ∅})
89 breq2 4689 . . . . . . . . . . . . 13 (𝑏 = ∅ → (𝑐𝑏𝑐 ≼ ∅))
9089rabbidv 3220 . . . . . . . . . . . 12 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
9190unieqd 4478 . . . . . . . . . . 11 (𝑏 = ∅ → {𝑐𝐴𝑐𝑏} = {𝑐𝐴𝑐 ≼ ∅})
9291eqeq2d 2661 . . . . . . . . . 10 (𝑏 = ∅ → (∅ = {𝑐𝐴𝑐𝑏} ↔ ∅ = {𝑐𝐴𝑐 ≼ ∅}))
9392rspcev 3340 . . . . . . . . 9 ((∅ ∈ ω ∧ ∅ = {𝑐𝐴𝑐 ≼ ∅}) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
9476, 88, 93sylancr 696 . . . . . . . 8 (( [] Or 𝐴𝐴 ⊆ Fin) → ∃𝑏 ∈ ω ∅ = {𝑐𝐴𝑐𝑏})
95 eqeq1 2655 . . . . . . . . 9 (𝑑 = ∅ → (𝑑 = {𝑐𝐴𝑐𝑏} ↔ ∅ = {𝑐𝐴𝑐𝑏}))
9695rexbidv 3081 . . . . . . . 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 3786 . . . 4 (𝑑 ∈ (𝐴 ∪ {∅}) ↔ (𝑑𝐴𝑑 ∈ {∅}))
102100, 101syl6bbr 278 . . 3 (( [] Or 𝐴𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏} ↔ 𝑑 ∈ (𝐴 ∪ {∅})))
103102abbi1dv 2772 . 2 (( [] Or 𝐴𝐴 ⊆ Fin) → {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = {𝑐𝐴𝑐𝑏}} = (𝐴 ∪ {∅}))
1042, 103syl5eq 2697 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 1523   ∈ wcel 2030  {cab 2637   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945   ∪ cun 3605   ⊆ wss 3607  ∅c0 3948  {csn 4210  ∪ cuni 4468   class class class wbr 4685   ↦ cmpt 4762   Or wor 5063  ran crn 5144  ‘cfv 5926   [⊊] crpss 6978  ωcom 7107   ≈ cen 7994   ≼ cdom 7995  Fincfn 7997  cardccrd 8799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-rpss 6979  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803 This theorem is referenced by:  fin1a2lem12  9271
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