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Theorem tskuni 10640

Description: The union of an element of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 22-Jun-2013.)

**Assertion**
Ref	Expression
tskuni	⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ∈ 𝑇)

Proof of Theorem tskuni Dummy variables 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsksdom 10613	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ 𝑇)
2		cardidg 10405	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ Tarski → (card‘𝑇) ≈ 𝑇)
3	2	ensymd 8866	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ Tarski → 𝑇 ≈ (card‘𝑇))
4	3	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝑇 ≈ (card‘𝑇))
5		sdomentr 8976	. . . . . . . . . . . 12 ⊢ ((𝐴 ≺ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → 𝐴 ≺ (card‘𝑇))
6	1, 4, 5	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ (card‘𝑇))
7		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥))
8	7	rnmpt 5896	. . . . . . . . . . . . . 14 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)}
9		cardon 9801	. . . . . . . . . . . . . . . . 17 ⊢ (card‘𝑇) ∈ On
10		sdomdom 8841	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ≺ (card‘𝑇) → 𝐴 ≼ (card‘𝑇))
11		ondomen 9894	. . . . . . . . . . . . . . . . 17 ⊢ (((card‘𝑇) ∈ On ∧ 𝐴 ≼ (card‘𝑇)) → 𝐴 ∈ dom card)
12	9, 10, 11	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ≺ (card‘𝑇) → 𝐴 ∈ dom card)
13	12	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → 𝐴 ∈ dom card)
14		vex 3445	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
15	14	imaex 7831	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ 𝑥) ∈ V
16	15, 7	fnmpti 6627	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) Fn 𝐴
17		dffn4 6745	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)))
18	16, 17	mpbi 229	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥))
19		fodomnum 9914	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom card → ((𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) → ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) ≼ 𝐴))
20	13, 18, 19	mpisyl 21	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) ≼ 𝐴)
21	8, 20	eqbrtrrid 5128	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≼ 𝐴)
22		domsdomtr 8977	. . . . . . . . . . . . 13 ⊢ (({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≼ 𝐴 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
23	21, 22	sylancom 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
24	23	adantll 711	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
25	6, 24	mpdan 684	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
26		ne0i 4281	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑇 → 𝑇 ≠ ∅)
27		tskcard 10638	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (card‘𝑇) ∈ Inacc)
28	26, 27	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (card‘𝑇) ∈ Inacc)
29		elina 10544	. . . . . . . . . . . 12 ⊢ ((card‘𝑇) ∈ Inacc ↔ ((card‘𝑇) ≠ ∅ ∧ (cf‘(card‘𝑇)) = (card‘𝑇) ∧ ∀𝑥 ∈ (card‘𝑇)𝒫 𝑥 ≺ (card‘𝑇)))
30	29	simp2bi 1145	. . . . . . . . . . 11 ⊢ ((card‘𝑇) ∈ Inacc → (cf‘(card‘𝑇)) = (card‘𝑇))
31	28, 30	syl 17	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (cf‘(card‘𝑇)) = (card‘𝑇))
32	25, 31	breqtrrd 5120	. . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
33	32	3adant2 1130	. . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
34	33	adantr 481	. . . . . . 7 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
35	28	3adant2 1130	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (card‘𝑇) ∈ Inacc)
36	35	adantr 481	. . . . . . . . 9 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (card‘𝑇) ∈ Inacc)
37		inawina 10547	. . . . . . . . 9 ⊢ ((card‘𝑇) ∈ Inacc → (card‘𝑇) ∈ Inacc_w)
38		winalim 10552	. . . . . . . . 9 ⊢ ((card‘𝑇) ∈ Inacc_w → Lim (card‘𝑇))
39	36, 37, 38	3syl 18	. . . . . . . 8 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → Lim (card‘𝑇))
40		vex 3445	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
41		eqeq1 2740	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑧 = (𝑓 “ 𝑥) ↔ 𝑦 = (𝑓 “ 𝑥)))
42	41	rexbidv 3171	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥)))
43	40, 42	elab 3619	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥))
44		imassrn 6010	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
45		f1ofo 6774	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → 𝑓:∪ 𝐴–onto→(card‘𝑇))
46		forn 6742	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → ran 𝑓 = (card‘𝑇))
47	45, 46	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ran 𝑓 = (card‘𝑇))
48	44, 47	sseqtrid 3984	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (𝑓 “ 𝑥) ⊆ (card‘𝑇))
49	48	ad2antlr 724	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ⊆ (card‘𝑇))
50		f1of1 6766	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → 𝑓:∪ 𝐴–1-1→(card‘𝑇))
51		elssuni 4885	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
52		vex 3445	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
53	52	f1imaen 8877	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:∪ 𝐴–1-1→(card‘𝑇) ∧ 𝑥 ⊆ ∪ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
54	50, 51, 53	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
55	54	adantll 711	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
56		simpl1 1190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑇 ∈ Tarski)
57		trss 5220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Tr 𝑇 → (𝐴 ∈ 𝑇 → 𝐴 ⊆ 𝑇))
58	57	imp 407	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)
59	58	3adant1 1129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)
60	59	sselda 3932	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑇)
61		tsksdom 10613	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → 𝑥 ≺ 𝑇)
62	56, 60, 61	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ 𝑇)
63	56, 3	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑇 ≈ (card‘𝑇))
64		sdomentr 8976	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≺ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → 𝑥 ≺ (card‘𝑇))
65	62, 63, 64	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ (card‘𝑇))
66	65	adantlr 712	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ (card‘𝑇))
67		ensdomtr 8978	. . . . . . . . . . . . . 14 ⊢ (((𝑓 “ 𝑥) ≈ 𝑥 ∧ 𝑥 ≺ (card‘𝑇)) → (𝑓 “ 𝑥) ≺ (card‘𝑇))
68	55, 66, 67	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≺ (card‘𝑇))
69	36, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (cf‘(card‘𝑇)) = (card‘𝑇))
70	69	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (cf‘(card‘𝑇)) = (card‘𝑇))
71	68, 70	breqtrrd 5120	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇)))
72		sseq1 3957	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ↔ (𝑓 “ 𝑥) ⊆ (card‘𝑇)))
73		breq1 5095	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓 “ 𝑥) → (𝑦 ≺ (cf‘(card‘𝑇)) ↔ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇))))
74	72, 73	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓 “ 𝑥) → ((𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇))) ↔ ((𝑓 “ 𝑥) ⊆ (card‘𝑇) ∧ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇)))))
75	74	biimprcd 249	. . . . . . . . . . . 12 ⊢ (((𝑓 “ 𝑥) ⊆ (card‘𝑇) ∧ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇))) → (𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))))
76	49, 71, 75	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))))
77	76	rexlimdva 3148	. . . . . . . . . 10 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))))
78	43, 77	biimtrid 241	. . . . . . . . 9 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} → (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))))
79	78	ralrimiv 3138	. . . . . . . 8 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → ∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇))))
80		fvex 6838	. . . . . . . . 9 ⊢ (card‘𝑇) ∈ V
81	80	cfslb2n 10125	. . . . . . . 8 ⊢ ((Lim (card‘𝑇) ∧ ∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≠ (card‘𝑇)))
82	39, 79, 81	syl2anc 584	. . . . . . 7 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≠ (card‘𝑇)))
83	34, 82	mpd 15	. . . . . 6 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≠ (card‘𝑇))
84	15	dfiun2 4980	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)}
85	48	ralrimivw 3143	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∀𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
86		iunss 4992	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇) ↔ ∀𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
87	85, 86	sylibr 233	. . . . . . . . 9 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
88		fof 6739	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → 𝑓:∪ 𝐴⟶(card‘𝑇))
89		foelrn 7038	. . . . . . . . . . . . 13 ⊢ ((𝑓:∪ 𝐴–onto→(card‘𝑇) ∧ 𝑦 ∈ (card‘𝑇)) → ∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧))
90	89	ex 413	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → ∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧)))
91		eluni2 4856	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝑥)
92		nfv 1916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥 𝑓:∪ 𝐴⟶(card‘𝑇)
93		nfiu1 4975	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)
94	93	nfel2 2922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)
95		ssiun2 4994	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → (𝑓 “ 𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
96	95	3ad2ant2 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓 “ 𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
97		ffn 6651	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → 𝑓 Fn ∪ 𝐴)
98	97	3ad2ant1 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑓 Fn ∪ 𝐴)
99	51	3ad2ant2 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ∪ 𝐴)
100		simp3 1137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥)
101		fnfvima 7165	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn ∪ 𝐴 ∧ 𝑥 ⊆ ∪ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑓 “ 𝑥))
102	98, 99, 100, 101	syl3anc 1370	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑓 “ 𝑥))
103	96, 102	sseldd 3933	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
104	103	3exp 1118	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))))
105	92, 94, 104	rexlimd 3245	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
106	91, 105	biimtrid 241	. . . . . . . . . . . . . 14 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑧 ∈ ∪ 𝐴 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
107		eleq1a 2832	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) → (𝑦 = (𝑓‘𝑧) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
108	106, 107	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑧 ∈ ∪ 𝐴 → (𝑦 = (𝑓‘𝑧) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))))
109	108	rexlimdv 3146	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
110	88, 90, 109	sylsyld 61	. . . . . . . . . . 11 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
111	45, 110	syl 17	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
112	111	ssrdv 3938	. . . . . . . . 9 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (card‘𝑇) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
113	87, 112	eqssd 3949	. . . . . . . 8 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) = (card‘𝑇))
114	84, 113	eqtr3id 2790	. . . . . . 7 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} = (card‘𝑇))
115	114	necon3ai 2965	. . . . . 6 ⊢ (∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≠ (card‘𝑇) → ¬ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
116	83, 115	syl 17	. . . . 5 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → ¬ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
117	116	pm2.01da 796	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
118	117	nexdv 1938	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
119		entr 8867	. . . . . . 7 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → ∪ 𝐴 ≈ (card‘𝑇))
120	3, 119	sylan2 593	. . . . . 6 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ∈ Tarski) → ∪ 𝐴 ≈ (card‘𝑇))
121		bren 8814	. . . . . 6 ⊢ (∪ 𝐴 ≈ (card‘𝑇) ↔ ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
122	120, 121	sylib 217	. . . . 5 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ∈ Tarski) → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
123	122	expcom 414	. . . 4 ⊢ (𝑇 ∈ Tarski → (∪ 𝐴 ≈ 𝑇 → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)))
124	123	3ad2ant1 1132	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (∪ 𝐴 ≈ 𝑇 → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)))
125	118, 124	mtod 197	. 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ ∪ 𝐴 ≈ 𝑇)
126		uniss 4860	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝑇 → ∪ 𝐴 ⊆ ∪ 𝑇)
127		df-tr 5210	. . . . . . . . . 10 ⊢ (Tr 𝑇 ↔ ∪ 𝑇 ⊆ 𝑇)
128	127	biimpi 215	. . . . . . . . 9 ⊢ (Tr 𝑇 → ∪ 𝑇 ⊆ 𝑇)
129	126, 128	sylan9ss 3945	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑇 ∧ Tr 𝑇) → ∪ 𝐴 ⊆ 𝑇)
130	129	expcom 414	. . . . . . 7 ⊢ (Tr 𝑇 → (𝐴 ⊆ 𝑇 → ∪ 𝐴 ⊆ 𝑇))
131	57, 130	syld 47	. . . . . 6 ⊢ (Tr 𝑇 → (𝐴 ∈ 𝑇 → ∪ 𝐴 ⊆ 𝑇))
132	131	imp 407	. . . . 5 ⊢ ((Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ⊆ 𝑇)
133		tsken 10611	. . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ ∪ 𝐴 ⊆ 𝑇) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
134	132, 133	sylan2 593	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ (Tr 𝑇 ∧ 𝐴 ∈ 𝑇)) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
135	134	3impb 1114	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
136	135	ord 861	. 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (¬ ∪ 𝐴 ≈ 𝑇 → ∪ 𝐴 ∈ 𝑇))
137	125, 136	mpd 15	1 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ∈ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∃wex 1780 ∈ wcel 2105 {cab 2713 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3898 ∅c0 4269 𝒫 cpw 4547 ∪ cuni 4852 ∪ ciun 4941 class class class wbr 5092 ↦ cmpt 5175 Tr wtr 5209 dom cdm 5620 ran crn 5621 “ cima 5623 Oncon0 6302 Lim wlim 6303 Fn wfn 6474 ⟶wf 6475 –1-1→wf1 6476 –onto→wfo 6477 –1-1-onto→wf1o 6478 ‘cfv 6479 ≈ cen 8801 ≼ cdom 8802 ≺ csdm 8803 cardccrd 9792 cfccf 9794 Inacc_wcwina 10539 Inacccina 10540 Tarskictsk 10605

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-inf2 9498 ax-ac2 10320

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-smo 8247 df-recs 8272 df-rdg 8311 df-1o 8367 df-2o 8368 df-er 8569 df-map 8688 df-ixp 8757 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-oi 9367 df-har 9414 df-r1 9621 df-card 9796 df-aleph 9797 df-cf 9798 df-acn 9799 df-ac 9973 df-wina 10541 df-ina 10542 df-tsk 10606

This theorem is referenced by: tskwun 10641 tskint 10642 tskun 10643 tskurn 10646 pwinfi3 41501

W3C validator