Theorem iundom2g 10500

Description: An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
iunfo.1	⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
iundomg.2	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴)
iundomg.3	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶)

Assertion

Ref	Expression
iundom2g	⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑇(𝑥)

Proof of Theorem iundom2g

Dummy variables 𝑓 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iundomg.2	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴)
2		iundomg.3	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶)
3		brdomi 8934	. . . . . . . . 9 ⊢ (𝐵 ≼ 𝐶 → ∃𝑔 𝑔:𝐵–1-1→𝐶)
4	3	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → ∃𝑔 𝑔:𝐵–1-1→𝐶)
5		f1f 6759	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐶 → 𝑔:𝐵⟶𝐶)
6		reldom 8927	. . . . . . . . . . . . . . 15 ⊢ Rel ≼
7	6	brrelex2i 5698	. . . . . . . . . . . . . 14 ⊢ (𝐵 ≼ 𝐶 → 𝐶 ∈ V)
8	6	brrelex1i 5697	. . . . . . . . . . . . . 14 ⊢ (𝐵 ≼ 𝐶 → 𝐵 ∈ V)
9	7, 8	elmapd 8816	. . . . . . . . . . . . 13 ⊢ (𝐵 ≼ 𝐶 → (𝑔 ∈ (𝐶 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐶))
10	9	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔 ∈ (𝐶 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐶))
11	5, 10	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔:𝐵–1-1→𝐶 → 𝑔 ∈ (𝐶 ↑m 𝐵)))
12		ssiun2 5014	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ↑m 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵))
13	12	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝐶 ↑m 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵))
14	13	sseld 3948	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔 ∈ (𝐶 ↑m 𝐵) → 𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)))
15	11, 14	syld 47	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔:𝐵–1-1→𝐶 → 𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)))
16	15	ancrd 551	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔:𝐵–1-1→𝐶 → (𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∧ 𝑔:𝐵–1-1→𝐶)))
17	16	eximdv 1917	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (∃𝑔 𝑔:𝐵–1-1→𝐶 → ∃𝑔(𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∧ 𝑔:𝐵–1-1→𝐶)))
18	4, 17	mpd 15	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → ∃𝑔(𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∧ 𝑔:𝐵–1-1→𝐶))
19		df-rex 3055	. . . . . . 7 ⊢ (∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:𝐵–1-1→𝐶 ↔ ∃𝑔(𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∧ 𝑔:𝐵–1-1→𝐶))
20	18, 19	sylibr 234	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:𝐵–1-1→𝐶)
21	20	ralimiaa 3066	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶 → ∀𝑥 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:𝐵–1-1→𝐶)
22	2, 21	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:𝐵–1-1→𝐶)
23		nfv 1914	. . . . 5 ⊢ Ⅎ𝑦∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:𝐵–1-1→𝐶
24		nfiu1 4994	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)
25		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑥𝑔
26		nfcsb1v 3889	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
27		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
28	25, 26, 27	nff1 6757	. . . . . 6 ⊢ Ⅎ𝑥 𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶
29	24, 28	nfrexw 3289	. . . . 5 ⊢ Ⅎ𝑥∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶
30		csbeq1a 3879	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
31		f1eq2 6755	. . . . . . 7 ⊢ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 → (𝑔:𝐵–1-1→𝐶 ↔ 𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
32	30, 31	syl 17	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑔:𝐵–1-1→𝐶 ↔ 𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
33	32	rexbidv 3158	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:𝐵–1-1→𝐶 ↔ ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
34	23, 29, 33	cbvralw 3282	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:𝐵–1-1→𝐶 ↔ ∀𝑦 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶)
35	22, 34	sylib 218	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶)
36		f1eq1 6754	. . . 4 ⊢ (𝑔 = (𝑓‘𝑦) → (𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
37	36	acni3 10007	. . 3 ⊢ ((∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶) → ∃𝑓(𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
38	1, 35, 37	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
39		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑦(𝑓‘𝑥):𝐵–1-1→𝐶
40		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑦)
41	40, 26, 27	nff1 6757	. . . . . 6 ⊢ Ⅎ𝑥(𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶
42		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
43		f1eq1 6754	. . . . . . . 8 ⊢ ((𝑓‘𝑥) = (𝑓‘𝑦) → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):𝐵–1-1→𝐶))
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):𝐵–1-1→𝐶))
45		f1eq2 6755	. . . . . . . 8 ⊢ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 → ((𝑓‘𝑦):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
46	30, 45	syl 17	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑦):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
47	44, 46	bitrd 279	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
48	39, 41, 47	cbvralw 3282	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶)
49		df-ne 2927	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
50		acnrcl 10002	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴 → 𝐴 ∈ V)
51	1, 50	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
52		r19.2z 4461	. . . . . . . . . . . 12 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 ≼ 𝐶)
53	7	rexlimivw 3131	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≼ 𝐶 → 𝐶 ∈ V)
54	52, 53	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) → 𝐶 ∈ V)
55	54	expcom 413	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶 → (𝐴 ≠ ∅ → 𝐶 ∈ V))
56	2, 55	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ≠ ∅ → 𝐶 ∈ V))
57		xpexg 7729	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
58	51, 56, 57	syl6an 684	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ≠ ∅ → (𝐴 × 𝐶) ∈ V))
59	49, 58	biimtrrid 243	. . . . . . 7 ⊢ (𝜑 → (¬ 𝐴 = ∅ → (𝐴 × 𝐶) ∈ V))
60		xpeq1 5655	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
61		0xp 5740	. . . . . . . . 9 ⊢ (∅ × 𝐶) = ∅
62		0ex 5265	. . . . . . . . 9 ⊢ ∅ ∈ V
63	61, 62	eqeltri 2825	. . . . . . . 8 ⊢ (∅ × 𝐶) ∈ V
64	60, 63	eqeltrdi 2837	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) ∈ V)
65	59, 64	pm2.61d2 181	. . . . . 6 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V)
66		iunfo.1	. . . . . . . . . 10 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
67	66	eleq2i 2821	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑇 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
68		eliun 4962	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵))
69	67, 68	bitri 275	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑇 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵))
70		r19.29 3095	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)) → ∃𝑥 ∈ 𝐴 ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵)))
71		xp1st 8003	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑥} × 𝐵) → (1st ‘𝑦) ∈ {𝑥})
72	71	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (1st ‘𝑦) ∈ {𝑥})
73		elsni 4609	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑦) ∈ {𝑥} → (1st ‘𝑦) = 𝑥)
74	72, 73	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (1st ‘𝑦) = 𝑥)
75		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → 𝑥 ∈ 𝐴)
76	74, 75	eqeltrd 2829	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (1st ‘𝑦) ∈ 𝐴)
77	74	fveq2d 6865	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (𝑓‘(1st ‘𝑦)) = (𝑓‘𝑥))
78	77	fveq1d 6863	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘𝑥)‘(2nd ‘𝑦)))
79		f1f 6759	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥):𝐵–1-1→𝐶 → (𝑓‘𝑥):𝐵⟶𝐶)
80	79	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (𝑓‘𝑥):𝐵⟶𝐶)
81		xp2nd 8004	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑦) ∈ 𝐵)
82	81	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (2nd ‘𝑦) ∈ 𝐵)
83	80, 82	ffvelcdmd 7060	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → ((𝑓‘𝑥)‘(2nd ‘𝑦)) ∈ 𝐶)
84	78, 83	eqeltrd 2829	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) ∈ 𝐶)
85	76, 84	opelxpd 5680	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶))
86	85	rexlimiva 3127	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶))
87	70, 86	syl 17	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)) → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶))
88	87	ex 412	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶)))
89	69, 88	biimtrid 242	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (𝑦 ∈ 𝑇 → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶)))
90		fvex 6874	. . . . . . . . . 10 ⊢ (1st ‘𝑦) ∈ V
91		fvex 6874	. . . . . . . . . 10 ⊢ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) ∈ V
92	90, 91	opth 5439	. . . . . . . . 9 ⊢ (⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ = ⟨(1st ‘𝑧), ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))⟩ ↔ ((1st ‘𝑦) = (1st ‘𝑧) ∧ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))))
93		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (1st ‘𝑦) = (1st ‘𝑧))
94	93	fveq2d 6865	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (𝑓‘(1st ‘𝑦)) = (𝑓‘(1st ‘𝑧)))
95	94	fveq1d 6863	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑧)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧)))
96	95	eqeq2d 2741	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑧)) ↔ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))))
97		djussxp 5812	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
98	66, 97	eqsstri 3996	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 ⊆ (𝐴 × V)
99		simprl 770	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → 𝑦 ∈ 𝑇)
100	98, 99	sselid 3947	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → 𝑦 ∈ (𝐴 × V))
101	100	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → 𝑦 ∈ (𝐴 × V))
102		xp1st 8003	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐴 × V) → (1st ‘𝑦) ∈ 𝐴)
103	101, 102	syl 17	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (1st ‘𝑦) ∈ 𝐴)
104		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶)
105		nfcv 2892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑓‘(1st ‘𝑦))
106		nfcsb1v 3889	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋(1st ‘𝑦) / 𝑥⦌𝐵
107	105, 106, 27	nff1 6757	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶
108		fveq2 6861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑦) → (𝑓‘𝑥) = (𝑓‘(1st ‘𝑦)))
109		f1eq1 6754	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) = (𝑓‘(1st ‘𝑦)) → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):𝐵–1-1→𝐶))
110	108, 109	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (1st ‘𝑦) → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):𝐵–1-1→𝐶))
111		csbeq1a 3879	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑦) → 𝐵 = ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
112		f1eq2 6755	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = ⦋(1st ‘𝑦) / 𝑥⦌𝐵 → ((𝑓‘(1st ‘𝑦)):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶))
113	111, 112	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (1st ‘𝑦) → ((𝑓‘(1st ‘𝑦)):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶))
114	110, 113	bitrd 279	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘𝑦) → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶))
115	107, 114	rspc 3579	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑦) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶))
116	103, 104, 115	sylc 65	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶)
117	106	nfel2 2911	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵
118	74	eqcomd 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → 𝑥 = (1st ‘𝑦))
119	118, 111	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → 𝐵 = ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
120	82, 119	eleqtrd 2831	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
121	120	ex 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 → (((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵))
122	117, 121	rexlimi 3238	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥 ∈ 𝐴 ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
123	70, 122	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
124	123	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵))
125	69, 124	biimtrid 242	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (𝑦 ∈ 𝑇 → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵))
126	125	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ 𝑇) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
127	126	adantrr 717	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
128	127	adantr 480	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
129	125	ralrimiv 3125	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → ∀𝑦 ∈ 𝑇 (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
130		fveq2 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (2nd ‘𝑦) = (2nd ‘𝑧))
131		fveq2 6861	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (1st ‘𝑦) = (1st ‘𝑧))
132	131	csbeq1d 3869	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → ⦋(1st ‘𝑦) / 𝑥⦌𝐵 = ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
133	130, 132	eleq12d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵 ↔ (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵))
134	133	rspccva 3590	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝑇 (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵 ∧ 𝑧 ∈ 𝑇) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
135	129, 134	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑧 ∈ 𝑇) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
136	135	adantrl 716	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
137	136	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
138	93	csbeq1d 3869	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → ⦋(1st ‘𝑦) / 𝑥⦌𝐵 = ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
139	137, 138	eleqtrrd 2832	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
140		f1fveq 7240	. . . . . . . . . . . . 13 ⊢ (((𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶 ∧ ((2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵 ∧ (2nd ‘𝑧) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)) → (((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑧)) ↔ (2nd ‘𝑦) = (2nd ‘𝑧)))
141	116, 128, 139, 140	syl12anc 836	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑧)) ↔ (2nd ‘𝑦) = (2nd ‘𝑧)))
142	96, 141	bitr3d 281	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧)) ↔ (2nd ‘𝑦) = (2nd ‘𝑧)))
143	142	pm5.32da 579	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (((1st ‘𝑦) = (1st ‘𝑧) ∧ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))) ↔ ((1st ‘𝑦) = (1st ‘𝑧) ∧ (2nd ‘𝑦) = (2nd ‘𝑧))))
144		simprr 772	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → 𝑧 ∈ 𝑇)
145	98, 144	sselid 3947	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → 𝑧 ∈ (𝐴 × V))
146		xpopth 8012	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐴 × V) ∧ 𝑧 ∈ (𝐴 × V)) → (((1st ‘𝑦) = (1st ‘𝑧) ∧ (2nd ‘𝑦) = (2nd ‘𝑧)) ↔ 𝑦 = 𝑧))
147	100, 145, 146	syl2anc 584	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (((1st ‘𝑦) = (1st ‘𝑧) ∧ (2nd ‘𝑦) = (2nd ‘𝑧)) ↔ 𝑦 = 𝑧))
148	143, 147	bitrd 279	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (((1st ‘𝑦) = (1st ‘𝑧) ∧ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))) ↔ 𝑦 = 𝑧))
149	92, 148	bitrid 283	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ = ⟨(1st ‘𝑧), ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))⟩ ↔ 𝑦 = 𝑧))
150	149	ex 412	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ = ⟨(1st ‘𝑧), ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))⟩ ↔ 𝑦 = 𝑧)))
151	89, 150	dom2d 8967	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → ((𝐴 × 𝐶) ∈ V → 𝑇 ≼ (𝐴 × 𝐶)))
152	65, 151	syl5com 31	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → 𝑇 ≼ (𝐴 × 𝐶)))
153	48, 152	biimtrrid 243	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶 → 𝑇 ≼ (𝐴 × 𝐶)))
154	153	adantld 490	. . 3 ⊢ (𝜑 → ((𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶) → 𝑇 ≼ (𝐴 × 𝐶)))
155	154	exlimdv 1933	. 2 ⊢ (𝜑 → (∃𝑓(𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶) → 𝑇 ≼ (𝐴 × 𝐶)))
156	38, 155	mpd 15	1 ⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3450  ⦋csb 3865   ⊆ wss 3917  ∅c0 4299  {csn 4592  ⟨cop 4598  ∪ ciun 4958   class class class wbr 5110   × cxp 5639  ⟶wf 6510  –1-1→wf1 6511  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970   ↑_m cmap 8802   ≼ cdom 8919  AC wacn 9898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-dom 8923  df-acn 9902

This theorem is referenced by:  iundomg  10501  iundom  10502