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Theorem fiiuncl 41696
Description: If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
fiiuncl.xph 𝑥𝜑
fiiuncl.b ((𝜑𝑥𝐴) → 𝐵𝐷)
fiiuncl.un ((𝜑𝑦𝐷𝑧𝐷) → (𝑦𝑧) ∈ 𝐷)
fiiuncl.a (𝜑𝐴 ∈ Fin)
fiiuncl.n0 (𝜑𝐴 ≠ ∅)
Assertion
Ref Expression
fiiuncl (𝜑 𝑥𝐴 𝐵𝐷)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐵,𝑧   𝑥,𝐷,𝑦,𝑧   𝜑,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑦,𝑧)   𝐵(𝑥)

Proof of Theorem fiiuncl
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fiiuncl.n0 . 2 (𝜑𝐴 ≠ ∅)
2 neeq1 3052 . . . 4 (𝑣 = ∅ → (𝑣 ≠ ∅ ↔ ∅ ≠ ∅))
3 iuneq1 4900 . . . . 5 (𝑣 = ∅ → 𝑥𝑣 𝐵 = 𝑥 ∈ ∅ 𝐵)
43eleq1d 2877 . . . 4 (𝑣 = ∅ → ( 𝑥𝑣 𝐵𝐷 𝑥 ∈ ∅ 𝐵𝐷))
52, 4imbi12d 348 . . 3 (𝑣 = ∅ → ((𝑣 ≠ ∅ → 𝑥𝑣 𝐵𝐷) ↔ (∅ ≠ ∅ → 𝑥 ∈ ∅ 𝐵𝐷)))
6 neeq1 3052 . . . 4 (𝑣 = 𝑤 → (𝑣 ≠ ∅ ↔ 𝑤 ≠ ∅))
7 iuneq1 4900 . . . . 5 (𝑣 = 𝑤 𝑥𝑣 𝐵 = 𝑥𝑤 𝐵)
87eleq1d 2877 . . . 4 (𝑣 = 𝑤 → ( 𝑥𝑣 𝐵𝐷 𝑥𝑤 𝐵𝐷))
96, 8imbi12d 348 . . 3 (𝑣 = 𝑤 → ((𝑣 ≠ ∅ → 𝑥𝑣 𝐵𝐷) ↔ (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷)))
10 neeq1 3052 . . . 4 (𝑣 = (𝑤 ∪ {𝑢}) → (𝑣 ≠ ∅ ↔ (𝑤 ∪ {𝑢}) ≠ ∅))
11 iuneq1 4900 . . . . 5 (𝑣 = (𝑤 ∪ {𝑢}) → 𝑥𝑣 𝐵 = 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵)
1211eleq1d 2877 . . . 4 (𝑣 = (𝑤 ∪ {𝑢}) → ( 𝑥𝑣 𝐵𝐷 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵𝐷))
1310, 12imbi12d 348 . . 3 (𝑣 = (𝑤 ∪ {𝑢}) → ((𝑣 ≠ ∅ → 𝑥𝑣 𝐵𝐷) ↔ ((𝑤 ∪ {𝑢}) ≠ ∅ → 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵𝐷)))
14 neeq1 3052 . . . 4 (𝑣 = 𝐴 → (𝑣 ≠ ∅ ↔ 𝐴 ≠ ∅))
15 iuneq1 4900 . . . . 5 (𝑣 = 𝐴 𝑥𝑣 𝐵 = 𝑥𝐴 𝐵)
1615eleq1d 2877 . . . 4 (𝑣 = 𝐴 → ( 𝑥𝑣 𝐵𝐷 𝑥𝐴 𝐵𝐷))
1714, 16imbi12d 348 . . 3 (𝑣 = 𝐴 → ((𝑣 ≠ ∅ → 𝑥𝑣 𝐵𝐷) ↔ (𝐴 ≠ ∅ → 𝑥𝐴 𝐵𝐷)))
18 neirr 2999 . . . . 5 ¬ ∅ ≠ ∅
1918pm2.21i 119 . . . 4 (∅ ≠ ∅ → 𝑥 ∈ ∅ 𝐵𝐷)
2019a1i 11 . . 3 (𝜑 → (∅ ≠ ∅ → 𝑥 ∈ ∅ 𝐵𝐷))
21 iunxun 4982 . . . . . . . 8 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 = ( 𝑥𝑤 𝐵 𝑥 ∈ {𝑢}𝐵)
22 nfcsb1v 3855 . . . . . . . . . 10 𝑥𝑢 / 𝑥𝐵
23 vex 3447 . . . . . . . . . 10 𝑢 ∈ V
24 csbeq1a 3845 . . . . . . . . . 10 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
2522, 23, 24iunxsnf 41695 . . . . . . . . 9 𝑥 ∈ {𝑢}𝐵 = 𝑢 / 𝑥𝐵
2625uneq2i 4090 . . . . . . . 8 ( 𝑥𝑤 𝐵 𝑥 ∈ {𝑢}𝐵) = ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵)
2721, 26eqtri 2824 . . . . . . 7 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 = ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵)
28 iuneq1 4900 . . . . . . . . . . . . . 14 (𝑤 = ∅ → 𝑥𝑤 𝐵 = 𝑥 ∈ ∅ 𝐵)
29 0iun 4952 . . . . . . . . . . . . . . 15 𝑥 ∈ ∅ 𝐵 = ∅
3029a1i 11 . . . . . . . . . . . . . 14 (𝑤 = ∅ → 𝑥 ∈ ∅ 𝐵 = ∅)
3128, 30eqtrd 2836 . . . . . . . . . . . . 13 (𝑤 = ∅ → 𝑥𝑤 𝐵 = ∅)
3231uneq1d 4092 . . . . . . . . . . . 12 (𝑤 = ∅ → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) = (∅ ∪ 𝑢 / 𝑥𝐵))
33 0un 4303 . . . . . . . . . . . . . 14 (∅ ∪ 𝑢 / 𝑥𝐵) = 𝑢 / 𝑥𝐵
34 unidm 4082 . . . . . . . . . . . . . 14 (𝑢 / 𝑥𝐵𝑢 / 𝑥𝐵) = 𝑢 / 𝑥𝐵
3533, 34eqtr4i 2827 . . . . . . . . . . . . 13 (∅ ∪ 𝑢 / 𝑥𝐵) = (𝑢 / 𝑥𝐵𝑢 / 𝑥𝐵)
3635a1i 11 . . . . . . . . . . . 12 (𝑤 = ∅ → (∅ ∪ 𝑢 / 𝑥𝐵) = (𝑢 / 𝑥𝐵𝑢 / 𝑥𝐵))
3732, 36eqtrd 2836 . . . . . . . . . . 11 (𝑤 = ∅ → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) = (𝑢 / 𝑥𝐵𝑢 / 𝑥𝐵))
3837adantl 485 . . . . . . . . . 10 (((𝜑𝑢 ∈ (𝐴𝑤)) ∧ 𝑤 = ∅) → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) = (𝑢 / 𝑥𝐵𝑢 / 𝑥𝐵))
39 simpl 486 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝐴𝑤)) → 𝜑)
40 eldifi 4057 . . . . . . . . . . . . 13 (𝑢 ∈ (𝐴𝑤) → 𝑢𝐴)
4140adantl 485 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝐴𝑤)) → 𝑢𝐴)
42 fiiuncl.xph . . . . . . . . . . . . . . . 16 𝑥𝜑
43 nfv 1915 . . . . . . . . . . . . . . . 16 𝑥 𝑢𝐴
4442, 43nfan 1900 . . . . . . . . . . . . . . 15 𝑥(𝜑𝑢𝐴)
45 nfcv 2958 . . . . . . . . . . . . . . . 16 𝑥𝐷
4622, 45nfel 2972 . . . . . . . . . . . . . . 15 𝑥𝑢 / 𝑥𝐵𝐷
4744, 46nfim 1897 . . . . . . . . . . . . . 14 𝑥((𝜑𝑢𝐴) → 𝑢 / 𝑥𝐵𝐷)
48 eleq1 2880 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑢 → (𝑥𝐴𝑢𝐴))
4948anbi2d 631 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → ((𝜑𝑥𝐴) ↔ (𝜑𝑢𝐴)))
5024eleq1d 2877 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → (𝐵𝐷𝑢 / 𝑥𝐵𝐷))
5149, 50imbi12d 348 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (((𝜑𝑥𝐴) → 𝐵𝐷) ↔ ((𝜑𝑢𝐴) → 𝑢 / 𝑥𝐵𝐷)))
52 fiiuncl.b . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐵𝐷)
5347, 51, 52chvarfv 2241 . . . . . . . . . . . . 13 ((𝜑𝑢𝐴) → 𝑢 / 𝑥𝐵𝐷)
5434, 53eqeltrid 2897 . . . . . . . . . . . 12 ((𝜑𝑢𝐴) → (𝑢 / 𝑥𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)
5539, 41, 54syl2anc 587 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (𝐴𝑤)) → (𝑢 / 𝑥𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)
5655adantr 484 . . . . . . . . . 10 (((𝜑𝑢 ∈ (𝐴𝑤)) ∧ 𝑤 = ∅) → (𝑢 / 𝑥𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)
5738, 56eqeltrd 2893 . . . . . . . . 9 (((𝜑𝑢 ∈ (𝐴𝑤)) ∧ 𝑤 = ∅) → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)
5857adantlr 714 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝐴𝑤)) ∧ (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷)) ∧ 𝑤 = ∅) → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)
59 simplll 774 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝐴𝑤)) ∧ (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷)) ∧ ¬ 𝑤 = ∅) → 𝜑)
6040ad3antlr 730 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝐴𝑤)) ∧ (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷)) ∧ ¬ 𝑤 = ∅) → 𝑢𝐴)
61 neqne 2998 . . . . . . . . . . . 12 𝑤 = ∅ → 𝑤 ≠ ∅)
6261adantl 485 . . . . . . . . . . 11 (((𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷) ∧ ¬ 𝑤 = ∅) → 𝑤 ≠ ∅)
63 simpl 486 . . . . . . . . . . 11 (((𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷) ∧ ¬ 𝑤 = ∅) → (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷))
6462, 63mpd 15 . . . . . . . . . 10 (((𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷) ∧ ¬ 𝑤 = ∅) → 𝑥𝑤 𝐵𝐷)
6564adantll 713 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝐴𝑤)) ∧ (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷)) ∧ ¬ 𝑤 = ∅) → 𝑥𝑤 𝐵𝐷)
66533adant3 1129 . . . . . . . . . 10 ((𝜑𝑢𝐴 𝑥𝑤 𝐵𝐷) → 𝑢 / 𝑥𝐵𝐷)
67 simp3 1135 . . . . . . . . . 10 ((𝜑𝑢𝐴 𝑥𝑤 𝐵𝐷) → 𝑥𝑤 𝐵𝐷)
68 simp1 1133 . . . . . . . . . . 11 ((𝜑𝑢𝐴 𝑥𝑤 𝐵𝐷) → 𝜑)
6968, 67, 663jca 1125 . . . . . . . . . 10 ((𝜑𝑢𝐴 𝑥𝑤 𝐵𝐷) → (𝜑 𝑥𝑤 𝐵𝐷𝑢 / 𝑥𝐵𝐷))
70 eleq1 2880 . . . . . . . . . . . . . 14 (𝑧 = 𝑢 / 𝑥𝐵 → (𝑧𝐷𝑢 / 𝑥𝐵𝐷))
71703anbi3d 1439 . . . . . . . . . . . . 13 (𝑧 = 𝑢 / 𝑥𝐵 → ((𝜑 𝑥𝑤 𝐵𝐷𝑧𝐷) ↔ (𝜑 𝑥𝑤 𝐵𝐷𝑢 / 𝑥𝐵𝐷)))
72 uneq2 4087 . . . . . . . . . . . . . 14 (𝑧 = 𝑢 / 𝑥𝐵 → ( 𝑥𝑤 𝐵𝑧) = ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵))
7372eleq1d 2877 . . . . . . . . . . . . 13 (𝑧 = 𝑢 / 𝑥𝐵 → (( 𝑥𝑤 𝐵𝑧) ∈ 𝐷 ↔ ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) ∈ 𝐷))
7471, 73imbi12d 348 . . . . . . . . . . . 12 (𝑧 = 𝑢 / 𝑥𝐵 → (((𝜑 𝑥𝑤 𝐵𝐷𝑧𝐷) → ( 𝑥𝑤 𝐵𝑧) ∈ 𝐷) ↔ ((𝜑 𝑥𝑤 𝐵𝐷𝑢 / 𝑥𝐵𝐷) → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)))
7574imbi2d 344 . . . . . . . . . . 11 (𝑧 = 𝑢 / 𝑥𝐵 → (( 𝑥𝑤 𝐵𝐷 → ((𝜑 𝑥𝑤 𝐵𝐷𝑧𝐷) → ( 𝑥𝑤 𝐵𝑧) ∈ 𝐷)) ↔ ( 𝑥𝑤 𝐵𝐷 → ((𝜑 𝑥𝑤 𝐵𝐷𝑢 / 𝑥𝐵𝐷) → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) ∈ 𝐷))))
76 eleq1 2880 . . . . . . . . . . . . . 14 (𝑦 = 𝑥𝑤 𝐵 → (𝑦𝐷 𝑥𝑤 𝐵𝐷))
77763anbi2d 1438 . . . . . . . . . . . . 13 (𝑦 = 𝑥𝑤 𝐵 → ((𝜑𝑦𝐷𝑧𝐷) ↔ (𝜑 𝑥𝑤 𝐵𝐷𝑧𝐷)))
78 uneq1 4086 . . . . . . . . . . . . . 14 (𝑦 = 𝑥𝑤 𝐵 → (𝑦𝑧) = ( 𝑥𝑤 𝐵𝑧))
7978eleq1d 2877 . . . . . . . . . . . . 13 (𝑦 = 𝑥𝑤 𝐵 → ((𝑦𝑧) ∈ 𝐷 ↔ ( 𝑥𝑤 𝐵𝑧) ∈ 𝐷))
8077, 79imbi12d 348 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑤 𝐵 → (((𝜑𝑦𝐷𝑧𝐷) → (𝑦𝑧) ∈ 𝐷) ↔ ((𝜑 𝑥𝑤 𝐵𝐷𝑧𝐷) → ( 𝑥𝑤 𝐵𝑧) ∈ 𝐷)))
81 fiiuncl.un . . . . . . . . . . . 12 ((𝜑𝑦𝐷𝑧𝐷) → (𝑦𝑧) ∈ 𝐷)
8280, 81vtoclg 3518 . . . . . . . . . . 11 ( 𝑥𝑤 𝐵𝐷 → ((𝜑 𝑥𝑤 𝐵𝐷𝑧𝐷) → ( 𝑥𝑤 𝐵𝑧) ∈ 𝐷))
8375, 82vtoclg 3518 . . . . . . . . . 10 (𝑢 / 𝑥𝐵𝐷 → ( 𝑥𝑤 𝐵𝐷 → ((𝜑 𝑥𝑤 𝐵𝐷𝑢 / 𝑥𝐵𝐷) → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)))
8466, 67, 69, 83syl3c 66 . . . . . . . . 9 ((𝜑𝑢𝐴 𝑥𝑤 𝐵𝐷) → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)
8559, 60, 65, 84syl3anc 1368 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝐴𝑤)) ∧ (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷)) ∧ ¬ 𝑤 = ∅) → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)
8658, 85pm2.61dan 812 . . . . . . 7 (((𝜑𝑢 ∈ (𝐴𝑤)) ∧ (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷)) → ( 𝑥𝑤 𝐵𝑢 / 𝑥𝐵) ∈ 𝐷)
8727, 86eqeltrid 2897 . . . . . 6 (((𝜑𝑢 ∈ (𝐴𝑤)) ∧ (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷)) → 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵𝐷)
8887a1d 25 . . . . 5 (((𝜑𝑢 ∈ (𝐴𝑤)) ∧ (𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷)) → ((𝑤 ∪ {𝑢}) ≠ ∅ → 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵𝐷))
8988ex 416 . . . 4 ((𝜑𝑢 ∈ (𝐴𝑤)) → ((𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷) → ((𝑤 ∪ {𝑢}) ≠ ∅ → 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵𝐷)))
9089adantrl 715 . . 3 ((𝜑 ∧ (𝑤𝐴𝑢 ∈ (𝐴𝑤))) → ((𝑤 ≠ ∅ → 𝑥𝑤 𝐵𝐷) → ((𝑤 ∪ {𝑢}) ≠ ∅ → 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵𝐷)))
91 fiiuncl.a . . 3 (𝜑𝐴 ∈ Fin)
925, 9, 13, 17, 20, 90, 91findcard2d 8748 . 2 (𝜑 → (𝐴 ≠ ∅ → 𝑥𝐴 𝐵𝐷))
931, 92mpd 15 1 (𝜑 𝑥𝐴 𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wnf 1785  wcel 2112  wne 2990  csb 3831  cdif 3881  cun 3882  wss 3884  c0 4246  {csn 4528   ciun 4884  Fincfn 8496
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-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-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-iun 4886  df-br 5034  df-opab 5096  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-om 7565  df-1o 8089  df-er 8276  df-en 8497  df-fin 8500
This theorem is referenced by:  fiunicl  41698  caragenfiiuncl  43151
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